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How to Solve Equations with Variables on Both Sides

Last Updated: March 11, 2023 Fact Checked

This article was co-authored by JohnK Wright V . JohnK Wright V is a Certified Math Teacher at Bridge Builder Academy in Plano, Texas. With over 20 years of teaching experience, he is a Texas SBEC Certified 8-12 Mathematics Teacher. He has taught in six different schools and has taught pre-algebra, algebra 1, geometry, algebra 2, pre-calculus, statistics, math reasoning, and math models with applications. He was a Mathematics Major at Southeastern Louisiana and he has a Bachelor of Science from The University of the State of New York (now Excelsior University) and a Master of Science in Computer Information Systems from Boston University. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 185,551 times.

To study algebra, you will see equations that have a variable on one side, but later on you will often see equations that have variables on both sides. The most important thing to remember when solving such equations is that whatever you do to one side of the equation, you must do to the other side. Using this rule, it is easy to move variables around so that you can isolate them and use basic operations to find their value.

Solving Equations with One Variable on Both Sides

Step 1 Apply the distributive property, if necessary.

Solving System Equations with Two Variables

Step 1 Isolate a variable in one equation.

Solving Example Problems

Step 1 Try this problem using the distributive property with one variable:

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  • ↑ https://www.khanacademy.org/math/cc-sixth-grade-math/cc-6th-expressions-and-variables/cc-6th-distributive-property/v/the-distributive-property
  • ↑ https://www.virtualnerd.com/algebra-1/linear-equations-solve/variables-both-sides-equations/variables-both-sides-solution/variables-grouping-symbols-both-sides
  • ↑ https://www.youtube.com/watch?v=hrAOSknrYiI&t=296s
  • ↑ https://www.khanacademy.org/math/cc-sixth-grade-math/cc-6th-expressions-and-variables/cc-6th-evaluating-expressions/v/expression-terms-factors-and-coefficients
  • ↑ https://www.virtualnerd.com/pre-algebra/linear-functions-graphing/system-of-equations/solving-systems-equations/two-equations-two-variables-substitution

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Module 9: Multi-Step Linear Equations

Solving equations with variables on both sides, learning outcomes.

  • Identify the constant and variable terms in an equation
  • Solve linear equations by isolating constants and variables
  • Solve linear equations with variables on both sides that require several steps

The equations we solved in the last section simplified nicely so that we could use the division property to isolate the variable and solve the equation. Sometimes, after you simplify you may have a variable and a constant term on the same side of the equal sign.

Our strategy will involve choosing one side of the equation to be the variable side, and the other side of the equation to be the constant side. This will help us with organization. Then, we will use the Subtraction and Addition Properties of Equality, step by step, to isolate the variable terms on one side of the equation.

Read on to find out how to solve this kind of equation.

Solve: [latex]4x+6=-14[/latex]

In this equation, the variable is only on the left side. It makes sense to call the left side the variable side. Therefore, the right side will be the constant side.

Solve: [latex]2y - 7=15[/latex]

Solution: Notice that the variable is only on the left side of the equation, so this will be the variable side and the right side will be the constant side. Since the left side is the variable side, the [latex]7[/latex] is out of place. It is subtracted from the [latex]2y[/latex], so to “undo” subtraction, add [latex]7[/latex] to both sides.

Now you can try a similar problem.

Solve Equations with Variables on Both Sides

You may have noticed that in all the equations we have solved so far, we had variables on only one side of the equation. This does not happen all the time—so now we’ll see how to solve equations where there are variable terms on both sides of the equation. We will start like we did above—choosing a variable side and a constant side, and then use the Subtraction and Addition Properties of Equality to collect all variables on one side and all constants on the other side. Remember, what you do to the left side of the equation, you must do to the right side as well.

In the next example, the variable, [latex]x[/latex], is on both sides, but the constants appear only on the right side, so we’ll make the right side the “constant” side. Then the left side will be the “variable” side.

Solve: [latex]5x=4x+7[/latex]

Solve: [latex]7x=-x+24[/latex].

Solution: The only constant, [latex]24[/latex], is on the right, so let the left side be the variable side.

Did you see the subtle difference between the two equations? In the first, the right side looked like this: [latex]4x+7[/latex], and in the second, the right side looked like this: [latex]-x+24[/latex], even though they look different, we still used the same techniques to solve both.

Now you can try solving an equation with variables on both sides where it is beneficial to move the variable term to the left side.

In our last examples, we moved the variable term to the left side of the equation. In the next example, you will see that it is beneficial to move the variable term to the right side of the equation. There is no “correct” side to move the variable term, but the choice can help you avoid working with negative signs.

Solve: [latex]5y - 8=7y[/latex]

Solution: The only constant, [latex]-8[/latex], is on the left side of the equation, and the variable, [latex]y[/latex], is on both sides. Let’s leave the constant on the left and collect the variables to the right.

Now you can try solving an equation where it is beneficial to move the variable term to the right side.

Solve Equations with Variables and Constants on Both Sides

The next example will be the first to have variables and constants on both sides of the equation. As we did before, we’ll collect the variable terms to one side and the constants to the other side. You will see that as the number of variable and constant terms increases, so do the number of steps it takes to solve the equation.

Solve: [latex]7x+5=6x+2[/latex]

Solution: Start by choosing which side will be the variable side and which side will be the constant side. The variable terms are [latex]7x[/latex] and [latex]6x[/latex]. Since [latex]7[/latex] is greater than [latex]6[/latex], make the left side the variable side and so the right side will be the constant side.

Solve: [latex]6n - 2=-3n+7[/latex]

In the following video we show an example of how to solve a multi-step equation by moving the variable terms to one side and the constants to the other side. You will see that it doesn’t matter which side you choose to be the variable side; you can get the correct answer either way.

In the next example, we move the variable terms to the right side to keep a positive coefficient on the variable.

Solve: [latex]2a - 7=5a+8[/latex]

This equation has [latex]2a[/latex] on the left and [latex]5a[/latex] on the right. Since [latex]5>2[/latex], make the right side the variable side and the left side the constant side.

The following video shows another example of solving a multi-step equation by moving the variable terms to one side and the constants to the other side.

Try these problems to see how well you understand how to solve linear equations with variables and constants on both sides of the equal sign.

We just showed a lot of examples of different kinds of linear equations you may encounter. There are some good habits to develop that will help you solve all kinds of linear equations. We’ll summarize the steps we took so you can easily refer to them.

Solve an equation with variables and constants on both sides

  • Choose one side to be the variable side and then the other will be the constant side.
  • Collect the variable terms to the variable side, using the Addition or Subtraction Property of Equality.
  • Collect the constants to the other side, using the Addition or Subtraction Property of Equality.
  • Make the coefficient of the variable [latex]1[/latex], using the Multiplication or Division Property of Equality.
  • Check the solution by substituting it into the original equation.
  • Question ID 142131, 142125, 142129, 142132, 142134, 142136. Authored by : Lumen Learning. License : CC BY: Attribution . License Terms : IMathAS Community License, CC-BY + GPL
  • Solve a Linear Equation in One Variable with Variables on Both Sides: 2m-9=6m-17. Authored by : James Sousa (Mathispower4u.com) for Lumen Learning. Located at : https://youtu.be/kiYPW6hrTS4 . License : CC BY: Attribution
  • Solve a Linear Equation in One Variable with Variables on Both Sides: 2x+8=-2x-24. Authored by : James Sousa (Mathispower4u.com) for Lumen Learning. Located at : https://youtu.be/_hBoWoctfAo . License : CC BY: Attribution
  • Ex: Solve an Equation with Variable Terms on Both Sides. Authored by : James Sousa (Mathispower4u.com). Located at : https://youtu.be/xfXGgqgJyDE . License : CC BY: Attribution
  • Prealgebra. Provided by : OpenStax. License : CC BY: Attribution . License Terms : Download for free at http://cnx.org/contents/[email protected]

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Calcworkshop

How to Solve Equations with Variables on Both Sides 25 Examples!

// Last Updated: January 20, 2020 - Watch Video //

This video is just an extension of our previous lesson on how to solve multi-step equations.

Jenn (B.S., M.Ed.) of Calcworkshop® teaching variables on both sides

Jenn, Founder Calcworkshop ® , 15+ Years Experience (Licensed & Certified Teacher)

Now that we know how to SCAM an equation, we are going to apply these same skills for solving linear equations with variables on both sides .

The trick to solving these types of equations is to know that if we can add or multiply a number to both sides of an equation, then we can do the same thing with variables!

Remember, a variable is a symbol or letter that represents one or more numbers.

So when we move a variable from one side of the equation to the other, we’re just moving a number! And that’s why we can do it!

Now, we know that our answer (solution or root), is one that makes the equation true.

But for some equations, there are no answers or roots, for we will end up with a false statement! When this happens, we call it the empty set or null set.

Using several transformations to solve an equation

Use the Distributive Property to Solve

And even more puzzling, is that sometimes our answer will be every real number, not just one!

In these instances, we say that our equation is an identity, as Khan Academy nicely states.

Together we will walk through countless examples, similar to the one on the left, to solve equations with variables on both sides.

In the end you will feel comfortable and confident in SCAMing even the most difficult of questions , just like the example you see to the left.

Variables on Both Sides – Video

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Still wondering if CalcWorkshop is right for you? Take a Tour and find out how a membership can take the struggle out of learning math.

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how to solve equations with variables on both sides with negative numbers

Introduction:

Solving equations with variables on both sides can be a bit tricky compared to linear equations with a variable on one side. However, with the right techniques and practice, finding the solution can become an easy task. In this article, we will explore three methods that can help you solve equations with variables on both sides effectively.

Method 1: Simplification by Combining Like Terms

The first method involves simplifying the equation by combining like terms on each side of the equation.

1. Combine all the terms containing variables on one side of the equation, and all constant terms (i.e., numbers) on the other side.

2. Once you have combined like terms, you should now have a simplified linear equation.

3. Solve for the variable using inverse operations or traditional algebraic techniques.

Example: Solve 2x + 4 = x – 1

Step 1: Combine like terms.

2x – x = -1 – 4

Step 2: Simplify

Method 2: Using the Distributive Property

Another method for solving equations with variables on both sides involves using the distributive property to either add or subtract terms in parentheses before solving for the variable.

1. Apply the distributive property to expand terms within parentheses.

2. Combine like terms after distributing.

Example: Solve x(3 – x) = 2x + 3

Step 1: Use the distributive property.

3x – x^2 = 2x + 3

Step 2: Combine like terms.

-x^2 + x – 3 = 0 (Move all terms to one side)

Step 3: Solve for the variable. (In this case, use factoring or quadratic formula)

Method 3: Using Substitution

In cases where you have two different variables on both sides, you might need to use substitution to solve the equation.

1. Isolate one variable in one of the equations (if there are multiple equations).

2. Substitute the expression for the isolated variable into the other equation.

3. Solve for the remaining variable in the substituted equation.

4. Finally, substitute the value of the found variable into either of the original equations to find the value of the other variable.

Example: Solve 2x + y = 6 and x – y = 4

Step 1: Isolate one variable (we will isolate “y” in equation 1)

y = 6 – 2x

Step 2: Substitute into equation 2

x – (6 – 2x) = 4

Step 3: Solve for x

Step 4: Substitute x back into either equation to find y

y = 6 – 2(10/3)

Conclusion:

These three methods provide a solid foundation in solving equations with variables on both sides. While each method may be more useful in specific situations, understanding and practicing all of them will equip you with a range of tools to tackle various types of algebraic problems. Keep practicing, and you will master these techniques in no time.

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how to solve equations with variables on both sides with negative numbers

Home / United States / Math Classes / Solving Equations with Variables on Boths Sides

Solving Equations with Variables on Boths Sides

Some equations have variables on both sides of the equation. In such cases, the equations have different types of soluti ons. Learn how to use the properties of the equations to determine the nature of the solution of the equation and to find the solution. ...Read More Read Less

About Equations of Variables on Both Sides

how to solve equations with variables on both sides with negative numbers

Solving equations with variables on both sides of the equation

  • Two sides of an equation

Solution of an equation

Solving an equation using properties of equalities, using the distributive property to solve an equation.

  • Equations with no solution

Equations with infinitely many solutions

  • Frequently Asked Questions

An equation is a mathematical statement that connects two equal expressions with an “equal to” sign. 2x  +  5  =  25  –  2x, y – 6  =  10 – 3y, and 8z  –  6  =  29 –  z are some examples of equations having variables on both sides. The unknown values like x, y, and z found in these equations are known as variables. We can find the values of variables of an equation by rearranging the terms using some special properties.

Two Sides of an Equation

Every equation has two sides: the expression on the left-hand side (LHS) of the equation and the expression on the right-hand side of the equation (RHS). In every true equation, the left-hand side will be equal to the right-hand side (LHS = RHS).

A solution of an equation is a value that can be used in the place of the variable to make a true statement. For example, the solution of x in the equation x – 2 = 0 is 2. If we use 2 in the place of x , we get (2) – 2 = 0 , which is a true statement. Some equations have only one solution. Certain equations have no solution. And certain equations have infinitely many solutions. 

While solving an equation, we are essentially finding the value of the unknown variable. So, the solution of an equation is the value of the variable. All equations can be solved by simplifying the terms on both sides and isolating the variable to find its solution. We need to use inverse math operations to isolate the variable. The general procedure for solving multi-step equations is as follows:

new1

First, we use the distributive property to expand the terms inside the parentheses. Then, we combine the like terms and simplify the equation. Finally, we isolate the variable using inverse operations. 

For example , let’s solve the equation y – 6 = 10 – 3y .

y – 6 = 10 – 3y                       Write the equation

Add 6 on both sides using the addition property of equality to simplify the equation. 

y – 6 + 6 = 10 – 3y + 6           Subtraction property of equality

y = 16 – 3y                             Subtract

Add 3y on both sides using the addition property of equality to simplify the equation. 

y + 3y = 16 – 3y + 3y           Subtraction property of equality

4y = 16                                   Subtract

Divide both sides by 4       using the division property of equality. 

\(\frac{4y}{4}\) = \(\frac{16}{4}\)                                 Division property of equality

y = 4                                     Simplify

So, the value of y in this equation is 4. 

The distributive property states that multiplying the sum of two or more addends by a number is the same as multiplying each addend individually by the number and then adding the products together. We can use distributive property to expand the terms inside the parentheses. 

Suppose you want to solve the equation 2 ( 2x + 5 ) = 25 – x . The first step is to expand the terms inside the parentheses using distributive property. 

2 ( 2x + 5 ) = 25 -x             Write the equation

4x + 10 = 25 – x                 Distributive property

Now, we combine the like terms together. We can add x on both sides. 

4x + 10 + x = 25 -x + x      

5x + 10 = 25                       Add

We can subtract 10 on both sides using the subtraction property of equality. 

5x + 10 – 10 = 25 – 10

5x = 15                               Subtract

Now, divide both sides by 5 using the division property of equality to isolate the variable.

\(\frac{5x}{5}\) = \(\frac{15}{5}\)                             Division property of equality

x = 3                                 Simplify

So, the solution of this equation is x = 3 .

Equations with no Solution

Unlike the examples we solved before, certain equations may not have any solution. When solving such equations, we will get an equivalent equation that is not a true statement for any value of the variable. For example, the final equation or statement might be 5 = 0 , which is not true. 

5x + 3 = 5x + 5 is an example of an equation that has no solution. 

The first step in solving the equation is to group the like terms. 

5x + 3 = 5x + 5                             Write the equation

5x + 3 – 5x = 5x + 5 – 5x            Subtraction property of equality

We write the final equation as 3 = 5 , which is not a true statement. Hence, the equation 5x + 3 = 5x + 5 does not have any solution. 

When solving equations with infinitely many solutions, we will get an equivalent solution that is true for all values of the variable. 8x + 3 = 8x + 3 is an example of an equation with infinitely many solutions. To solve this equation, we can group the common terms. 

8x + 3 = 8x + 3                 Write the equation

-8x   -8x                           Subtraction property of equality

The solution 3 = 3 is true for all values of the variable. Hence, the equation has infinitely many solutions. 

Example 1: Solve 5 ( x – 1 ) =  -4

5 ( x – 1 ) =  -4                   Write the equation

5x – 5 = -4                         Distributive property

5x – 5 + 5 = -4 + 5           Addition property of equality

5x = 1                                 Add

\(\frac{5x}{5}\) = \(\frac{1}{5}\)                              Division property of equality

\(x=\frac{1}{5}\)                               Simplify

Therefore, the solution of the equation is \(x=\frac{1}{5}\) .

Example 2: Solve 4 ( x – 3 ) = 4x – 12 .

4 ( x – 3 ) = 4x – 12                     Write the equation

4x – 12 = 4x – 12                         Distributive property

4x – 12  + 12 = 4x – 12  + 12       Addition property of equality

4x = 4x                                       Add

The equation has an infinite number of solutions as it is true for all values of x .

Example 3: Solve 8x – 3 = 8x + 5 .

8x – 3 = 8x + 5                       Write the equation

8x – 3 – 8x = 8x + 5 – 8x     Subtraction property of equality

-3 = 5                                     Subtract

The final equation is not a true statement. Hence, the equation has no solution. 

Example 4: A train starts its journey from New York with 250 passengers, stops at Philadelphia, and then reaches its destination at Washington. The number of passengers who boarded the train at Philadelphia is 3 times the number of passengers who got down at Philadelphia. Find the number of passengers who boarded the train from Philadelphia if 320 passengers got down at the final destination.

Number of passengers at the beginning of the journey  = 250

Let the number of passengers who got down at Philadelphia be x .

So, the number of passengers who boarded from Philadelphia is 3x .

Number of passengers who got down at the final destination = 320

An equation relating the number of passengers can be formed as follows:

250 – x + 3x = 320

Here, we are subtracting x from 250 because x number of passengers got down at Philadelphia. At the same time, 3x passengers boarded the train from the same station. So, we add 3x to the left hand side of the equation. 

250 – x + 3x = 320                       Write the equation

250 + 2x = 320                           Group the like terms

250 + 2x – 250 = 320 – 250       Subtraction property of equality

2x = 70                                         Subtract

\(\frac{2x}{2}\)   = \(\frac{70}{2}\)                                        Division property of equality

x = 35                                           Simplify

The solution of the equation 250 – x + 3x = 320  is 35.

So, 35 passengers got down at Philadelphia, and 3 x 35 = 105 passengers boarded the train from the same station. 

What is the difference between an equation and an expression?

An expression is a mathematical phrase that contains numbers and variables. Expressions will not have an “equal to” sign. An equation is a mathematical statement that connects two equal expressions with an “equal to” sign

Can a linear equation have 3 solutions?

No, a linear equation has either a unique solution, no solution, or infinitely many solutions. 

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EVERYTHING YOU NEED FOR THE YEAR >>> ALL ACCESS

Maneuvering the Middle

Student-Centered Math Lessons

Solving Equations in Middle School Math

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how to solve equations with variables on both sides with negative numbers

Solving equations is foundational for middle and high school math. Students actually have been doing this since first grade! However, students can make careless errors or struggle when it comes to following the many procedural steps required to solve an equation.

how to solve equations with variables on both sides with negative numbers

The Standard

how to solve equations with variables on both sides with negative numbers

Why the struggle?

Solving equations is very revealing. If your students struggle with integer operations , then it will show up again when solving equations. If your students struggle with rational number operations, then it will show up again when solving equations.

We asked our Facebook group what students mostly struggle with, and here are some of the responses:

  • Combining like terms incorrectly
  • Dividing/multiplying a negative coefficient. Students trying to add instead.
  • Not distributing when they should. Example: 5 – 2(x + 2) = 10 simplifying to 3(x +2) = 10  
  • Students completing inverse operations on the same side of the equal sign instead of combining like terms (see below for picture)

how to solve equations with variables on both sides with negative numbers

Does this sound familiar?

how to solve equations with variables on both sides with negative numbers

Use Algebra Tiles (No, really, use them!)

I was resistant! Manipulatives can cause undue stress (tiny items + 30 children); it is a lot! However, I think that if I had used algebra tiles in my classroom, students would have been way more engaged ! I occasionally drew a model to demonstrate solving equations, but then Noelle showed me all the ways algebra tiles could be used to combine like terms, distribute, and solve two-step equations (even quadratics). I was on board!

Before using algebra tiles to introduce students to solving equations, use algebra tiles to demonstrate combining like terms. Combining like terms can be challenging for students, but when you ask students to combine all of the long green tiles (x), the long red tiles (-x), the small yellow tiles (+1), and the small red tiles (-1), it provides excellent concrete practice. It will also assist with zero pair understanding, which will be essential to solving equations with algebra tiles.

What would you rather combine as a student?  The terms or the algebra tiles?

Solving equations is foundational for middle and high school math. Students can struggle to complete the many procedural steps required. Teach students the conceptual knowledge necessary using algebra tiles! | maneuveringthemiddle.com

Let’s move on to solving equations. Look at the following example:

Solving equations is foundational for middle and high school math. Students can struggle to complete the many procedural steps required. Teach students the conceptual knowledge necessary using algebra tiles! | maneuveringthemiddle.com

Before exposing your students to two-step equations with variables on both sides, let them practice with one-step equations first. Scaffolding is key. I chose this example to show you how versatile algebra tiles are (variables on both sides and negative values)

To isolate the variable, students would need to recognize that they needed to get rid of positive 2.  To do this, they would add two negative tiles to make a zero pair. Adding negative 2 to one side would mean they would add to the other side to keep the equation balanced.

Solving equations is foundational for middle and high school math. Students can struggle to complete the many procedural steps required. Teach students the conceptual knowledge necessary using algebra tiles! | maneuveringthemiddle.com

After students have removed the two zero pairs on the left side of the equation, they might be a little stuck. They are exploring, after all. Remind students that we want all variables on one side, since we want to find out what one x is equal to. Students would then add a negative x to both sides to create another zero pair.

Solving equations is foundational for middle and high school math. Students can struggle to complete the many procedural steps required. Teach students the conceptual knowledge necessary using algebra tiles! | maneuveringthemiddle.com

Lastly, to get the solution for one x, students would divide the remaining red tiles among the two x tiles: X = -4.

Solving equations is foundational for middle and high school math. Students can struggle to complete the many procedural steps required. Teach students the conceptual knowledge necessary using algebra tiles! | maneuveringthemiddle.com

Students are actually solving for x by undoing the problem, by using inverse operations (adding -2 to both sides and dividing by 2), and by keeping the equation balanced (they are adding tiles to both sides). All the procedural steps that might mean nothing to students in a traditional problem have meaning when students have been exposed to practicing with algebra tiles.

Remember that algebra tiles (like most manipulatives) exist to make the math visual. They provide conceptual understanding. Eventually, students will move into the algorithm. When students are exploring, make sure all of the solutions are integers (you can’t break the tiles into pieces).

To learn more about students making the jump from concrete to abstract, please check out our posts about the CRA Framework – Part 1 and Part 2 . 

Another tip I recommend is to have students write out what is happening as they are using algebra tiles to solve an equation. If they are combining 3 green tiles with two red tiles, then they would need to write 3x-2x with evidence of only 1x remaining.

If you don’t have access to algebra tiles, students can use this website .

Solving equations is foundational for middle and high school math. Students can struggle to complete the many procedural steps required. Teach students the conceptual knowledge necessary using algebra tiles! | maneuveringthemiddle.com

Be sure to grab our Getting Started with Algebra Tiles freebie to learn more about using algebra tiles to tackle simplifying expressions, the distributive property, solving linear equations, adding and subtracting polynomials, multiplying and dividing polynomials, and factoring polynomials.

how to solve equations with variables on both sides with negative numbers

Helpful Tips

These teacher tips from our Math Teacher VIP Facebook Group might help your students.

  • Draw a line to separate the two sides of the equation.
  • Do Undo Line – this is another strategy that can help students.
  • Color-coding to help with combining like terms
  • Making sure to actually say (and make students say), “2 times x equals 5” as opposed to “2x = 5.”

For more resources, check out these units and bundles.

Equations and Inequalities Unit 6th Grade TEKS

What additional tips do you have? Do you use algebra tiles in your classroom?

Solving equations is foundational for middle and high school math. Students can struggle to complete the many procedural steps required. Teach students the conceptual knowledge necessary using algebra tiles! | maneuveringthemiddle.com

Getting Started with Algebra Tiles

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how to solve equations with variables on both sides with negative numbers

how to solve equations with variables on both sides with negative numbers

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Solving equations

Here you will learn about solving equations, including linear and quadratic algebraic equations, and how to solve them.

Students will first learn about solving equations in grade 8 as a part of expressions and equations, and again in high school as a part of reasoning with equations and inequalities.

What is solving an equation?

Solving equations is a step-by-step process to find the value of the variable. A variable is the unknown part of an equation, either on the left or right side of the equals sign. Sometimes, you need to solve multi-step equations which contain algebraic expressions.

To do this, you must use the order of operations, which is a systematic approach to equation solving. When you use the order of operations, you first solve any part of an equation located within parentheses. An equation is a mathematical expression that contains an equals sign.

For example,

\begin{aligned}y+6&=11\\\\ 3(x-3)&=12\\\\ \cfrac{2x+2}{4}&=\cfrac{x-3}{3}\\\\ 2x^{2}+3&x-2=0\end{aligned}

There are two sides to an equation, with the left side being equal to the right side. Equations will often involve algebra and contain unknowns, or variables, which you often represent with letters such as x or y.

You can solve simple equations and more complicated equations to work out the value of these unknowns. They could involve fractions, decimals or integers.

What is solving an equation?

Common Core State Standards

How does this relate to 8 th grade and high school math?

  • Grade 8 – Expressions and Equations (8.EE.C.7) Solve linear equations in one variable.
  • High school – Reasoning with Equations and Inequalities (HSA.REI.B.3) Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

[FREE] Math Equations Check for Understanding Quiz (Grade 6 to 8)

[FREE] Math Equations Check for Understanding Quiz (Grade 6 to 8)

Use this quiz to check your grade 6 to 8 students’ understanding of math equations. 10+ questions with answers covering a range of 6th, 7th and 8th grade math equations topics to identify areas of strength and support!

How to solve equations

In order to solve equations, you need to work out the value of the unknown variable by adding, subtracting, multiplying or dividing both sides of the equation by the same value.

  • Combine like terms .
  • Simplify the equation by using the opposite operation to both sides.
  • Isolate the variable on one side of the equation.

Solving equations examples

Example 1: solve equations involving like terms.

Solve for x.

Combine like terms.

Combine the q terms on the left side of the equation. To do this, subtract 4q from both sides.

The goal is to simplify the equation by combining like terms. Subtracting 4q from both sides helps achieve this.

After you combine like terms, you are left with q=9-4q.

2 Simplify the equation by using the opposite operation on both sides.

Add 4q to both sides to isolate q to one side of the equation.

The objective is to have all the q terms on one side. Adding 4q to both sides accomplishes this.

After you move the variable to one side of the equation, you are left with 5q=9.

3 Isolate the variable on one side of the equation.

Divide both sides of the equation by 5 to solve for q.

Dividing by 5 allows you to isolate q to one side of the equation in order to find the solution. After dividing both sides of the equation by 5, you are left with q=1 \cfrac{4}{5} \, .

Example 2: solve equations with variables on both sides

Combine the v terms on the same side of the equation. To do this, add 8v to both sides.

7v+8v=8-8v+8v

After combining like terms, you are left with the equation 15v=8.

Simplify the equation by using the opposite operation on both sides and isolate the variable to one side.

Divide both sides of the equation by 15 to solve for v. This step will isolate v to one side of the equation and allow you to solve.

15v \div 15=8 \div 15

The final solution to the equation 7v=8-8v is \cfrac{8}{15} \, .

Example 3: solve equations with the distributive property

Combine like terms by using the distributive property.

The 3 outside the parentheses needs to be multiplied by both terms inside the parentheses. This is called the distributive property.

\begin{aligned}& 3 \times c=3 c \\\\ & 3 \times(-5)=-15 \\\\ &3 c-15-4=2\end{aligned}

Once the 3 is distributed on the left side, rewrite the equation and combine like terms. In this case, the like terms are the constants on the left, –15 and –4. Subtract –4 from –15 to get –19.

Simplify the equation by using the opposite operation on both sides.

The goal is to isolate the variable, c, on one side of the equation. By adding 19 to both sides, you move the constant term to the other side.

\begin{aligned}& 3 c-19+19=2+19 \\\\ & 3 c=21\end{aligned}

Isolate the variable to one side of the equation.

To solve for c, you want to get c by itself.

Dividing both sides by 3 accomplishes this.

On the left side, \cfrac{3c}{3} simplifies to c, and on the right, \cfrac{21}{3} simplifies to 7.

The final solution is c=7.

As an additional step, you can plug 7 back into the original equation to check your work.

Example 4: solve linear equations

Combine like terms by simplifying.

Using steps to solve, you know that the goal is to isolate x to one side of the equation. In order to do this, you must begin by subtracting from both sides of the equation.

\begin{aligned} & 2x+5=15 \\\\ & 2x+5-5=15-5 \\\\ & 2x=10 \end{aligned}

Continue to simplify the equation by using the opposite operation on both sides.

Continuing with steps to solve, you must divide both sides of the equation by 2 to isolate x to one side.

\begin{aligned} & 2x \div 2=10 \div 2 \\\\ & x= 5 \end{aligned}

Isolate the variable to one side of the equation and check your work.

Plugging in 5 for x in the original equation and making sure both sides are equal is an easy way to check your work. If the equation is not equal, you must check your steps.

\begin{aligned}& 2(5)+5=15 \\\\ & 10+5=15 \\\\ & 15=15\end{aligned}

Example 5: solve equations by factoring

Solve the following equation by factoring.

Combine like terms by factoring the equation by grouping.

Multiply the coefficient of the quadratic term by the constant term.

2 x (-20) = -40

Look for two numbers that multiply to give you –40 and add up to the coefficient of 3. In this case, the numbers are 8 and –5 because 8 x -5=–40, and 8+–5=3.

Split the middle term using those two numbers, 8 and –5. Rewrite the middle term using the numbers 8 and –5.

2x^2+8x-5x-20=0

Group the terms in pairs and factor out the common factors.

2x^2+8x-5x-20=2x(x + 4)-5(x+4)=0

Now, you’ve factored the equation and are left with the following simpler equations 2x-5 and x+4.

This step relies on understanding the zero product property, which states that if two numbers multiply to give zero, then at least one of those numbers must equal zero.

Let’s relate this back to the factored equation (2x-5)(x+4)=0

Because of this property, either (2x-5)=0 or (x+4)=0

Isolate the variable for each equation and solve.

When solving these simpler equations, remember that you must apply each step to both sides of the equation to maintain balance.

\begin{aligned}& 2 x-5=0 \\\\ & 2 x-5+5=0+5 \\\\ & 2 x=5 \\\\ & 2 x \div 2=5 \div 2 \\\\ & x=\cfrac{5}{2} \end{aligned}

\begin{aligned}& x+4=0 \\\\ & x+4-4=0-4 \\\\ & x=-4\end{aligned}

The solution to this equation is x=\cfrac{5}{2} and x=-4.

Example 6: solve quadratic equations

Solve the following quadratic equation.

Combine like terms by factoring the quadratic equation when terms are isolated to one side.

To factorize a quadratic expression like this, you need to find two numbers that multiply to give -5 (the constant term) and add to give +2 (the coefficient of the x term).

The two numbers that satisfy this are -1 and +5.

So you can split the middle term 2x into -1x+5x: x^2-1x+5x-5-1x+5x

Now you can take out common factors x(x-1)+5(x-1).

And since you have a common factor of (x-1), you can simplify to (x+5)(x-1).

The numbers -1 and 5 allow you to split the middle term into two terms that give you common factors, allowing you to simplify into the form (x+5)(x-1).

Let’s relate this back to the factored equation (x+5)(x-1)=0.

Because of this property, either (x+5)=0 or (x-1)=0.

Now, you can solve the simple equations resulting from the zero product property.

\begin{aligned}& x+5=0 \\\\ & x+5-5=0-5 \\\\ & x=-5 \\\\\\ & x-1=0 \\\\ & x-1+1=0+1 \\\\ & x=1\end{aligned}

The solutions to this quadratic equation are x=1 and x=-5.

Teaching tips for solving equations

  • Use physical manipulatives like balance scales as a visual aid. Show how you need to keep both sides of the equation balanced, like a scale. Add or subtract the same thing from both sides to keep it balanced when solving. Use this method to practice various types of equations.
  • Emphasize the importance of undoing steps to isolate the variable. If you are solving for x and 3 is added to x, subtracting 3 undoes that step and isolates the variable x.
  • Relate equations to real-world, relevant examples for students. For example, word problems about tickets for sports games, cell phone plans, pizza parties, etc. can make the concepts click better.
  • Allow time for peer teaching and collaborative problem solving. Having students explain concepts to each other, work through examples on whiteboards, etc. reinforces the process and allows peers to ask clarifying questions. This type of scaffolding would be beneficial for all students, especially English-Language Learners. Provide supervision and feedback during the peer interactions.

Easy mistakes to make

  • Forgetting to distribute or combine like terms One common mistake is neglecting to distribute a number across parentheses or combine like terms before isolating the variable. This error can lead to an incorrect simplified form of the equation.
  • Misapplying the distributive property Incorrectly distributing a number across terms inside parentheses can result in errors. Students may forget to multiply each term within the parentheses by the distributing number, leading to an inaccurate equation.
  • Failing to perform the same operation on both sides It’s crucial to perform the same operation on both sides of the equation to maintain balance. Forgetting this can result in an imbalanced equation and incorrect solutions.
  • Making calculation errors Simple arithmetic mistakes, such as addition, subtraction, multiplication, or division errors, can occur during the solution process. Checking calculations is essential to avoid errors that may propagate through the steps.
  • Ignoring fractions or misapplying operations When fractions are involved, students may forget to multiply or divide by the common denominator to eliminate them. Misapplying operations on fractions can lead to incorrect solutions or complications in the final answer.

Related math equations lessons

  • Math equations
  • Rearranging equations
  • How to find the equation of a line
  • Solve equations with fractions
  • Linear equations
  • Writing linear equations
  • Substitution
  • Identity math
  • One step equation

Practice solving equations questions

1. Solve 4x-2=14.

GCSE Quiz False

Add 2 to both sides.

Divide both sides by 4.

2. Solve 3x-8=x+6.

Add 8 to both sides.

Subtract x from both sides.

Divide both sides by 2.

3. Solve 3(x+3)=2(x-2).

Expanding the parentheses.

Subtract 9 from both sides.

Subtract 2x from both sides.

4. Solve \cfrac{2 x+2}{3}=\cfrac{x-3}{2}.

Multiply by 6 (the lowest common denominator) and simplify.

Expand the parentheses.

Subtract 4 from both sides.

Subtract 3x from both sides.

5. Solve \cfrac{3 x^{2}}{2}=24.

Multiply both sides by 2.

Divide both sides by 3.

Square root both sides.

6. Solve by factoring:

Use factoring to find simpler equations.

Set each set of parentheses equal to zero and solve.

x=3 or x=10

Solving equations FAQs

The first step in solving a simple linear equation is to simplify both sides by combining like terms. This involves adding or subtracting terms to isolate the variable on one side of the equation.

Performing the same operation on both sides of the equation maintains the equality. This ensures that any change made to one side is also made to the other, keeping the equation balanced and preserving the solutions.

To handle variables on both sides of the equation, start by combining like terms on each side. Then, move all terms involving the variable to one side by adding or subtracting, and simplify to isolate the variable. Finally, perform any necessary operations to solve for the variable.

To deal with fractions in an equation, aim to eliminate them by multiplying both sides of the equation by the least common denominator. This helps simplify the equation and make it easier to isolate the variable. Afterward, proceed with the regular steps of solving the equation.

The next lessons are

  • Inequalities
  • Types of graph
  • Coordinate plane

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how to solve equations with variables on both sides with negative numbers

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A complete, free online christian homeschool curriculum for your family and mine, solve equations with variables on both sides.

In this lesson, you will learn to solve equations that have variables on both sides. There are many real world problems that can be solved by this type of equation. To solve this type of equation, you will have to get the terms with variables on one side of the equal sign. Once again, the goal is to isolate the variable on one side of the equation. It is very important in this type of equation that you show each step in the solution to avoid confusion. Are you ready to get started?

Example 1:   Remember – group the terms with variables on one side of the equation and simplify.

Check your solution: 60 – 3(5) = 9(5); 60 – 15 = 45

Example 2:    It works with negative numbers, too!

Check your solution: -4(36) + 72 = -2(36); -144 + 72 = -72

Investigat e

Jane enjoys playing tennis and wants to join a Tennis Club. Members at the Daves Creek Tennis Club pay $250 plus $5 per visit to play at the indoor courts. Non-members must pay $15 per visit. How many visits must a member make to the courts for it to cost her the same as non-paying members?

Summarize the problem, using only key words or phrases and rewrite the problem as a statement.

Jane will pay the same amount as a member or non-member after v trips to the tennis court.

Now, translate this information into an equation with variables on both sides and solve.

Interpret the solution: If Jane joins the Tennis Club, she will have to visit the indoor tennis court 25 times for it to cost the same as a non-member.

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Solving Equations With Variables On Both Sides Worksheets

Created: February 13, 2024

Last updated: February 13, 2024

For kids to learn how to solve an equation with variables on both sides, they need to use the multiplication table and do this mentally. The children should also learn to identify solutions to equations using both provided numbers and inverse operations. Now, all of this sounds like a complex concept, but you can break it down into simpler concepts using the solving equations with variables on both sides worksheets. 

Benefits Of Solving Multi Step Equations With Variables On Both Sides Worksheets

Teaching equations can get complicated very fast, and you need your kids to catch up in the short time the curriculum offers you. Using the solving equations with variables on both sides word problems worksheets can ease your teaching process as the kids have enough practice questions to work with.

Solving linear equations worksheets with variables on both sides helps kids understand the concept of equations much better. More than being a practice object, the kids can use the solving multi step equations with variables on both sides worksheets PDF as a resource on how to represent equations in real-life scenarios.

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Download Free Printable Worksheets Solving Equations With Variables On Both Sides

Some teachers may want to create pre algebra solving equations with variables on both sides worksheets by yourself; however, it can be limiting as you are one person. But, with math platforms, you can find solving equations with variables on both sides worksheets PDF with colorful objects and fun characters that can keep kids engaged and interested. When using algebra 1 solving equations with variables on both sides worksheets are fun, kids learn better.

As a seasoned educator with a Bachelor’s in Secondary Education and over three years of experience, I specialize in making mathematics accessible to students of all backgrounds through Brighterly. My expertise extends beyond teaching; I blog about innovative educational strategies and have a keen interest in child psychology and curriculum development. My approach is shaped by a belief in practical, real-life application of math, making learning both impactful and enjoyable.

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2.3: Solve Equations with Variables and Constants on Both Sides

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Learning Objectives

By the end of this section, you will be able to:

  • Solve an equation with constants on both sides
  • Solve an equation with variables on both sides
  • Solve an equation with variables and constants on both sides

Before you get started, take this readiness quiz.

  • Simplify: 4y−9+9. If you missed this problem, review Exercise 1.10.20 .

Solve Equations with Constants on Both Sides

In all the equations we have solved so far, all the variable terms were on only one side of the equation with the constants on the other side. This does not happen all the time—so now we will learn to solve equations in which the variable terms, or constant terms, or both are on both sides of the equation.

Our strategy will involve choosing one side of the equation to be the “variable side”, and the other side of the equation to be the “constant side.” Then, we will use the Subtraction and Addition Properties of Equality to get all the variable terms together on one side of the equation and the constant terms together on the other side.

By doing this, we will transform the equation that began with variables and constants on both sides into the form \(ax=b\). We already know how to solve equations of this form by using the Division or Multiplication Properties of Equality.

Example \(\PageIndex{1}\)

Solve: \(7x+8=−13\).

In this equation, the variable is found only on the left side. It makes sense to call the left side the “variable” side. Therefore, the right side will be the “constant” side. We will write the labels above the equation to help us remember what goes where.

This figure shows the equation 7x plus 8 equals negative 13, with the left side of the equation labeled “variable”, written in red, and the right side of the equation labeled “constant”, written in red.

Since the left side is the “xx”, or variable side, the 8 is out of place. We must “undo” adding 8 by subtracting 8, and to keep the equality we must subtract 8 from both sides.

Try It \(\PageIndex{2}\)

Solve: \(3x+4=−8\).

\(x=−4\)

Try It \(\PageIndex{3}\)

Solve: \(5a+3=−37\).

\(a=−8\)

Example \(\PageIndex{4}\)

Solve: \(8y−9=31\).

Notice, the variable is only on the left side of the equation, so we will call this side the “variable” side, and the right side will be the “constant” side. Since the left side is the “variable” side, the 9 is out of place. It is subtracted from the 8y, so to “undo” subtraction, add 9 to both sides. Remember, whatever you do to the left, you must do to the right.

Try It \(\PageIndex{5}\)

Solve: \(5y−9=16\).

Try It \(\PageIndex{6}\)

Solve: \(3m−8=19\).

Solve Equations with Variables and Constants on Both Sides

The next example will be the first to have variables and constants on both sides of the equation. It may take several steps to solve this equation, so we need a clear and organized strategy.

Example \(\PageIndex{7}\)

Solve: \(9x=8x−6\).

Here the variable is on both sides, but the constants only appear on the right side, so let’s make the right side the “constant” side. Then the left side will be the “variable” side.

Try It \(\PageIndex{8}\)

Solve: \(6n=5n−10\).

\(n = -10\)

Try It \(\PageIndex{9}\)

Solve: \(-6c = -7c - 1\)

Example \(\PageIndex{10}\)

Solve: \(5y - 9 = 8y\)

The only constant is on the left and the y’s are on both sides. Let’s leave the constant on the left and get the variables to the right.

Try It \(\PageIndex{11}\)

Solve: \(3p−14=5p\).

Try It \(\PageIndex{12}\)

Solve: \(8m + 9 = 5m\)

Example \(\PageIndex{13}\)

Solve: \(12x = -x + 26\)

The only constant is on the right, so let the left side be the “variable” side.

Try It \(\PageIndex{14}\)

Solve: \(12j = -4j + 32\)

Try It \(\PageIndex{15}\)

Solve: \(8h = -4h + 12\)

Example \(\PageIndex{16}\): How to Solve Equations with Variables and Constants on Both Sides

Solve: \(7x + 5 = 6x + 2\)

This figure is a table that has three columns and four rows. The first column is a header column, and it contains the names and numbers of each step. The second column contains further written instructions. The third column contains math. On the top row of the table, the first cell on the left reads: “Step 1. Choose which side will the “variable” side—the other side will be the “constant” side.” The text in the second cell reads: “The variable terms are 7 x and 6 x. Since 7 is greater than 6, we will make the left side the “x” side and so the right side will be the “constant” side.” The third cell contains the equation 7 x plus 5 equals 6 x plus 2, and the left side of the equation is labeled “variable” written in red, and the right side of the equation is labeled “constant” written in red.

Try It \(\PageIndex{17}\)

Solve: \(12x+8=6x+2\).

\(x=−1\)

Try It \(\PageIndex{18}\)

Solve: \(9y+4=7y+12\).

​​​​​​ We’ll list the steps below so you can easily refer to them. But we’ll call this the ‘Beginning Strategy’ because we’ll be adding some steps later in this chapter.

BEGINNING STRATEGY FOR SOLVING EQUATIONS WITH VARIABLES AND CONSTANTS ON BOTH SIDES OF THE EQUATION.

  • Choose which side will be the “variable” side—the other side will be the “constant” side.
  • Collect the variable terms to the “variable” side of the equation, using the Addition or Subtraction Property of Equality.
  • Collect all the constants to the other side of the equation, using the Addition or Subtraction Property of Equality.
  • Make the coefficient of the variable equal 1, using the Multiplication or Division Property of Equality.
  • Check the solution by substituting it into the original equation.

In Step 1, a helpful approach is to make the “variable” side the side that has the variable with the larger coefficient. This usually makes the arithmetic easier.

Example \(\PageIndex{19}\)

Solve: \(8n−4=−2n+6\).

In the first step, choose the variable side by comparing the coefficients of the variables on each side.

Try It \(\PageIndex{20}\)

Solve: \(8q - 5 = -4q + 7\)

Try It \(\PageIndex{21}\)

Solve: \(7n - 3 = n + 3\)

Example \(\PageIndex{22}\)

Solve: \(7a -3 = 13a + 7\)

Since 13>7, make the right side the “variable” side and the left side the “constant” side.

Try It \(\PageIndex{23}\)

Solve: \(2a - 2 = 6a + 18\)

Try It \(\PageIndex{24}\)

Solve: \(4k -1 = 7k + 17\)

In the last example, we could have made the left side the “variable” side, but it would have led to a negative coefficient on the variable term. (Try it!) While we could work with the negative, there is less chance of errors when working with positives. The strategy outlined above helps avoid the negatives!

To solve an equation with fractions, we just follow the steps of our strategy to get the solution!

Example \(\PageIndex{25}\)

Solve: \(\frac{4}{5}x + 6 = \frac{1}{4}x - 2\)

Since \(\frac{5}{4} > \frac{1}{4}\), make the left side the “variable” side and the right side the “constant” side.

Try It \(\PageIndex{26}\)

Solve: \(\frac{7}{8}x - 12 = -\frac{1}{8}x - 2\)

Try It \(\PageIndex{27}\)

Solve: \(\frac{7}{6}x + 11 = \frac{1}{6}y + 8\)

We will use the same strategy to find the solution for an equation with decimals.

Example \(\PageIndex{28}\)

Solve: \(7.8x+4=5.4x−8\).

Since \(7.8>5.4\), make the left side the “variable” side and the right side the “constant” side.

Try It \(\PageIndex{29}\)

Solve: \(2.8x + 12 = -1.4x - 9\)

Try It \(\PageIndex{30}\)

Solve: \(3.6y + 8 = 1.2y - 4\)

Key Concepts

IMAGES

  1. Variables on Both Sides

    how to solve equations with variables on both sides with negative numbers

  2. Solving Equation with variables on both sides of the equation

    how to solve equations with variables on both sides with negative numbers

  3. 3 Ways to Solve Equations with Variables on Both Sides

    how to solve equations with variables on both sides with negative numbers

  4. 3 Ways to Solve Equations with Variables on Both Sides

    how to solve equations with variables on both sides with negative numbers

  5. How to Solve Equations with Variables on Both Sides: 7 Steps

    how to solve equations with variables on both sides with negative numbers

  6. How to Solve Equations with Variables on Both Sides: 7 Steps

    how to solve equations with variables on both sides with negative numbers

VIDEO

  1. 3-4 Solve Equations with Variables on Both Sides

  2. Solve Equations w/Variables on Both Sides

  3. Algebra 1 Notes 2-4 Solve Equations Variables Both Sides Part 1

  4. Solve Equation Variables on Both Sides

  5. Solve equations with variables on both sides

  6. SOLVE Equations with VARIABLES on BOTH SIDES

COMMENTS

  1. Intro to equations with variables on both sides

    3*1 - 3*2x = 4-6x And multiply the terms 3-6x = 4-6x Then add 6x to both sides. 3-6x+6x = 4-6x+6x The -6x+6x becomes zero 3-0 = 4-0 And subtract the 0 from each sides 3=4

  2. 3 Ways to Solve Equations with Variables on Both Sides

    1 Apply the distributive property, if necessary. The distributive property states that Failed to parse (MathML if possible (experimental): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): a (b+c)=ab+ac . [1]

  3. Inequalities with variables on both sides (video)

    Dividing or multiplying by a negative reverses the relationship between the numbers which is why you need to reverse the inequality.

  4. 8.4: Solve Equations with Variables and Constants on Both Sides)

    Solve an Equation with Constants on Both Sides. You may have noticed that in all the equations we have solved so far, all the variable terms were on only one side of the equation with the constants on the other side. This does not happen all the time—so now we'll see how to solve equations where the variable terms and/or constant terms are on both sides of the equation.

  5. 8.5: Solve Equations with Variables and Constants on Both Sides (Part 2)

    Prealgebra 1e (OpenStax) 8: Solving Linear Equations

  6. Solving Equations With Variables on Both Sides

    In the first, the right side looked like this: [latex]4x+7[/latex], and in the second, the right side looked like this: [latex]-x+24[/latex], even though they look different, we still used the same techniques to solve both. Now you can try solving an equation with variables on both sides where it is beneficial to move the variable term to the ...

  7. 8.4: Solve Equations with Variables and Constants on Both Sides (Part 1)

    Example 8.4.3: Solve: 5x = 4x + 7. Solution. Here the variable, x, is on both sides, but the constants appear only on the right side, so let's make the right side the "constant" side. Then the left side will be the "variable" side. We don't want any variables on the right, so subtract the 4x. 5x− 4x = 4x− 4x + 7.

  8. Equations with Variables on Both Sides

    For a complete lesson on solving equations with variables on both sides, go to https://www.MathHelp.com - 1000+ online math lessons featuring a personal math teacher inside every lesson! In...

  9. Solving an equation with variables on both side and one solution

    Brian McLogan 👉 Learn how to solve multi-step equations with variable on both sides of the equation. An equation is a statement stating that two values are equal. A multi...

  10. Solving a two step equation with negative variable

    👉 Learn how to solve two step linear equations. A linear equation is an equation whose highest exponent on its variable(s) is 1. To solve for a variable in ...

  11. Solving Linear Equations with Negative Numbers

    World Languages. Solve a linear equation that has negative numbers and a variable. This video focuses on using inverse operations to solve for a variable. This video was submitted through the Innovation Math Challenge, a contest open to professional and nonprofessional producers.

  12. How to Solve Equations with Variables on Both Sides

    The trick to solving these types of equations is to know that if we can add or multiply a number to both sides of an equation, then we can do the same thing with variables! Why? Remember, a variable is a symbol or letter that represents one or more numbers. So when we move a variable from one side of the equation to the other, we're just ...

  13. 3 Ways to Solve Equations with Variables on Both Sides

    Spread the loveIntroduction: Solving equations with variables on both sides can be a bit tricky compared to linear equations with a variable on one side. However, with the right techniques and practice, finding the solution can become an easy task. In this article, we will explore three methods that can help you solve equations with variables on both sides effectively. Method 1: Simplification ...

  14. 9.3: Solve Equations with Variables and Constants on Both Sides

    In Step 1, a helpful approach is to make the "variable" side the side that has the variable with the larger coefficient. This usually makes the arithmetic easier. Example 9.3.19. Solve: 8n − 4 = − 2n + 6. Solution. In the first step, choose the variable side by comparing the coefficients of the variables on each side.

  15. Equation Calculator

    Sign in Free equations calculator - solve linear, quadratic, polynomial, radical, exponential and logarithmic equations with all the steps. Type in any equation to get the solution, steps and graph

  16. How to Solve Equations with Variables on Both Sides? (Examples)

    An equation is a mathematical statement that connects two equal expressions with an "equal to" sign. 2x + 5 = 25 - 2x, y - 6 = 10 - 3y, and 8z - 6 = 29 - z are some examples of equations having variables on both sides. The unknown values like x, y, and z found in these equations are known as variables.

  17. Equation with variables on both sides: fractions

    To solve the equation (3/4)x + 2 = (3/8)x - 4, we first eliminate fractions by multiplying both sides by the least common multiple of the denominators. Then, we add or subtract terms from both sides of the equation to group the x-terms on one side and the constants on the other. Finally, we solve and check as normal.

  18. Solving Equations in Middle School Math

    Remind students that we want all variables on one side, since we want to find out what one x is equal to. Students would then add a negative x to both sides to create another zero pair. Lastly, to get the solution for one x, students would divide the remaining red tiles among the two x tiles: X = -4.

  19. Solve equations with variables on both sides

    Well, we can't leave this equation without being solved. So let's go ahead and solve this two separate suasion. So we're now gonna add seven to both sides because negative seven plus 7 to 0 thes terms canceled so we're left with two backs equals, while 13 plus seven is 20. So to get X by itself, we're going to divide both sides of our equation ...

  20. Solving Equations

    3 Isolate the variable on one side of the equation. Divide both sides of the equation by 5 to solve for q. 5q \div 5 = 9 \div 5 . Dividing by 5 allows you to isolate q to one side of the equation in order to find the solution. After dividing both sides of the equation by 5, you are left with q=1 \cfrac{4}{5} \, .

  21. Page 2 of 4

    Now that all the variables are on the same side, it's a much simpler equation to solve. 12 = 5x - 8 means that if you multiply x by 5 and then subtract 8, the answer should be 12. We need to solve this by working backwards and undoing the subtraction first. We can undo subtracting 8 by adding 8 to both sides. Last, we can undo multiplying by 5 ...

  22. Solve Equations with Variables on Both Sides

    In this lesson, you will learn to solve equations that have variables on both sides. There are many real world problems that can be solved by this type of equation. To solve this type of equation, you will have to get the terms with variables on one side of the equal sign. Once again, the goal is to isolate the variable on one side of the equation.

  23. 3.4: Solving Equations with Variables and Constants on Both Sides

    Check the solution by substituting it into the original equation. In Step 1, a helpful approach is to make the "variable" side the side that has the variable with the larger coefficient. This usually makes the arithmetic easier. Solve: 8n − 4 = −2n + 6. Since 8 > − 2, make the left side the "variable" side.

  24. Solving Equations With Variables On Both Sides Worksheets

    Solving linear equations worksheets with variables on both sides helps kids understand the concept of equations much better. More than being a practice object, the kids can use the solving multi step equations with variables on both sides worksheets PDF as a resource on how to represent equations in real-life scenarios.

  25. 2.3: Solve Equations with Variables and Constants on Both Sides

    Example 2.3. 19. Solve: 8 n − 4 = − 2 n + 6. Solution. In the first step, choose the variable side by comparing the coefficients of the variables on each side. Since 8 > − 2, make the left side the "variable" side. We don't want variable terms on the right side—add 2n to both sides to leave only constants on the right.