3.8: Solve Equations with a Variable on Both Sides
The band is selling popcorn for a fundraiser. For the past few weeks, the students have been out there taking orders with the hope of raising enough money for new uniforms.
“I really hope that we make enough to get the blue ones,” Josie said to Jake and Karen at lunch.
“Me too,” Karen said.
“I was sick a lot of the time, so I didn’t sell as many boxes as I’d hoped to,” Jake sighed.
“That’s okay Jake. Those things happen,” Karen said smiling.
That afternoon after practice, Karen went to sort through the orders that had come in. She began counting all of the sales that the students had made. She discovered that she and Josie had sold the same amount of boxes. Josie sold thirtysix more boxes than Jake after all, Jake had been sick. She sold three times as many as Jake did. Karen began to figure out how many boxes Jake had sold.
Can you figure this out? You will need to understand how to work with variables in a new way to write an equation and solve it. You will learn how to do this in the following Concept.
Guidance
Do you remember how to solve a basic equation?
Consider the problem,
The strategy for solving this equation is to use inverse operations to isolate the variable,
What if you needed to solve an equation like this?
How do we solve an equation with variables on both sides of the equation?
To solve an equation that has the same variable on both sides of it, you will use the same basic strategy you already know. You will use inverse operations to isolate the variables on one side of the equation. You will do this by using inverse operations to get all the terms that include variables on one side of the equation and using inverse operations to get all the numerical terms on the other side. Once you do this, you will be able to solve for the variable.
Think about it logically and it makes perfect sense. You get the variables together on one side of the equation, and then you get the numbers together on the other side of the equation. Once you have done this, you can combine like terms and solve for the value of the variable.
Solve for
The variable,
Alternatively, we could subtract
Now, the only variable is on the right side of the equation. So, let's get all the numerical terms on the left side of the equation. Since 30 is added to
Now, we can use inverse operations to get the
The value of
Sometimes, an equation will have a set of parentheses and variables on both sides of the equation. The distributive property is very helpful in solving these equations.
Solve for
Our first step should be to simplify the expression on the right side of the equation. According the order of operations, we should combine the like terms inside the parentheses first. Then we can simplify the rest of that expression, like this:
Now, we notice that the variable,
Now, the only term with a variable,
The value of
Example A
Solution:
Example B
Solution:
Example C
Solution:
Now let's go back to the dilemma from the beginning of the Concept.
First, we write an equation.
Now we solve it for
Jake sold 18 boxes of popcorn.
Karen and Josie sold the same amount. We can use Karen’s information that she sold three times as many boxes as Jake did.
Josie and Karen each sold 54 boxes of popcorn.
Vocabulary
 Distributive Property
 states that we can simplify an expression with parentheses by multiply a term outside of the parentheses with each of the terms inside the parentheses.
 Inverse Operation
 the opposite operation
Guided Practice
Here is one for you to try on your own.
Let’s break down working on this problem. First, we need to move the terms with variables to the same side of the equation. Let’s move the
Here we performed the inverse operation and then simplified the equation. Now we can solve this just as we would any other two step equation. Take a look and be sure to watch out if you end up working with negative numbers. Don’t mix up the signs!
The value of
Video Review
Khan Academy Solving Linear Equations 3
Practice
Directions: Solve each equation with variables on both sides.

6x=2x+16 
5y=3y+12 
4y=y−18 
8x=10x+20 
7x=4x+24 
9y=2y−21 
−6x+22=5x 
15y=9y+36 
14x=10x−40 
19y=4y−30 
18x=2x−32 
4x+1=2x+5 
6x+4=4x+10 
8x+3=5x+9 
10y−4=6y−12 
8x−5=10x−13 
12y−8=14y+14 
18x−5=20x+19 
−20y+8=−8y−4
Directions: Solve each equation with variables on both sides, by simplifying each equation first by using the distributive property.

2(x+3)=8x 
3(x+5)=−2x 
9y=4(y−5)
distributive property
The distributive property states that the product of an expression and a sum is equal to the sum of the products of the expression and each term in the sum. For example, .Inverse Operation
Inverse operations are operations that "undo" each other. Multiplication is the inverse operation of division. Addition is the inverse operation of subtraction.Variable
A variable is a symbol used to represent an unknown or changing quantity. The most common variables are a, b, x, y, m, and n.Image Attributions
Description
Learning Objectives
Here you'll solve equations with a variable on both sides of the equation.