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Equations with Variables on Both Sides

Move all variables to one side and solve.

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Solve Equations with a Variable on Both Sides
License: CC BY-NC 3.0

The hockey team is selling chocolate bars for a fundraiser. For the past few weeks, the students have been out there taking orders with the hope of raising enough money for the big tournament. Karen was in charge of sorting 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 thirty-six more boxes than Jake. She sold three times as many as Jake did. So how many boxed did Jake sell?

In this concept, you will learn to solve equations with a variable on both sides of the equation.

Equations with a Variable on Both Sides

To solve an equation that has the same variable on both sides of it, you need to get the variables together on one side of the equation, and then get the numbers together on the other side of the equation.

Let's look at an example.

12+t=30+3t

Start by subtracting 3t and 12 from both sides of the equation.

12+t1212+t3t==30+3t303t12

 
Next, you can combine like terms.

1212+t3t2t==303t1218

Then, solve for the value of the variable by dividing both sides by -2.

2t2t2t===181829

The answer is -9.

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.

Let’s look at an example.

Solve for ‘a’:

 4a+16=13a(2a+3a)

First, 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.

4a+164a+164a+164a+16====13a(2a+3a)13a(5a)13a5a8a

Next, notice that the variable, a, is on both sides of the equation. You can use inverse operations to get all of the terms with the variable, a, on one side of the equation. Since there is a number on the left side of the equation and there is no number on the right side of the equation, it is easier to try to get all of the variable terms on the right side of the equation. You can get all of the variable terms on the right side of the equation by subtracting 4a from both sides.

4a+164a4a+1616===8a8a4a4a

Then, to solve for ‘a’, you divide both sides of the equation by 4.

16164a===4a4a44

The answer is 4.

Examples

Example 1

Earlier, you were given a problem about Karen and the chocolate bar orders. She knew the following:

  1. She sold as many chocolate bars as Josie.
  2. Josie sold thirty-six more boxes than Jake.
  3. She sold three times as many as Jake did.

Karen wanted to find out how many chocolate bars Jake sold.

First, you need to write an equation. Let x= the number of chocolate bars that Jake sold. Therefore, Josie sold (36+x)boxes of bars and Karen sold 3xboxes of bars. Therefore the equation is:

 x+36=3x

Next subtract x from both sides of the equation.

x+36xx+3636===3x3xx2x

Then, divide both sides by 2 to solve for ‘x’.

36362x===2x2x218

The answer is 18.

Jake sold 18 boxes of chocolate bars. This also means that Karen sold 3(18) or 54 boxes of chocolate bars. As well, Josie sold (18+36) or 54 boxes of chocolate bars.

Example 2

Solve for ‘x’ in the equation:

 6x+1=8x+3

First, you need to move the terms with variables to the same side of the equation. Let’s move the 6x. You can do this by using an inverse operation. You subtract 6x from both sides of the equation.

6x+16x6x+11===8x+38x6x+32x+3

Next, subtract 3 from both sides of the equation so that only the variable remains on the left hand side.

1132===2x+32x+332x

Then, divide both sides by 2.

222x===2x2x21The answer is -1.

Example 3

6x+3=9x+6

First, you need to move the terms with variables to the same side of the equation. You can subtract 6x from both sides of the equation.

6x+36x6x+33===9x+69x6x+63x+6

Next, subtract 6 from both sides of the equation so that only the variable remains on the left hand side.

3363===3x+63x+663x

Then, divide both sides by 3.

333x===3x3x31

The answer is -1.

Example 4

 4x+x+2=10x13 

First, you should combine like terms on the left side of the equation.

4x+x+25x+2==10x1310x13

Next, you need to move the terms with variables to the same side of the equation. You subtract 5x from both sides of the equation.

5x+25x5x+22===10x1310x5x135x13

Next, add 13 from both sides of the equation so that only the variable remains on the left hand side.

22+1315===5x135x13+135x

Then, divide both sides by 5.

15155x===5x5x33

The answer is 3.

Example 5

 8y+2y=20y+10

First, you should combine like terms on the left side of the equation.

8y+2y10y==20y+1020y+10

Next, you need to move the terms with variables to the same side of the equation. You subtract 20y from both sides of the equation.

10y10y20y10y===20y+1020y20y+1010

Then, divide both sides by -10.

10y10y10y===1010101

The answer is -1.

Review

Solve each equation with variables on both sides.

1. 6x=2x+16

2. 5y=3y+12

3. 4y=y18

4. 8x=10x+20

5. \begin{align*}7x = 4x + 24\end{align*}7x=4x+24

6. \begin{align*}9y = 2y− 21\end{align*}9y=2y21

7. \begin{align*}−6x + 22 = 5x\end{align*}6x+22=5x

8. \begin{align*}15y = 9y + 36\end{align*}15y=9y+36

9. \begin{align*}14x = 10x− 40\end{align*}14x=10x40

10. \begin{align*}19y = 4y− 30\end{align*}19y=4y30

11. \begin{align*}18x = 2x− 32\end{align*}18x=2x32

12. \begin{align*}4x + 1 = 2x + 5\end{align*}4x+1=2x+5

13. \begin{align*}6x + 4 = 4x + 10\end{align*}6x+4=4x+10

14. \begin{align*}8x + 3 = 5x + 9\end{align*}8x+3=5x+9

15. \begin{align*}10y− 4 = 6y− 12\end{align*}10y4=6y12

16. \begin{align*}8x− 5 = 10x− 13\end{align*}8x5=10x13

17. \begin{align*}12y− 8 = 14y + 14\end{align*}12y8=14y+14

18. \begin{align*}18x− 5 = 20x + 19\end{align*}18x5=20x+19

19. \begin{align*}−20y + 8 = −8y− 4\end{align*}20y+8=8y4

Solve each equation with variables on both sides, by simplifying each equation first by using the distributive property.

20. \begin{align*}2(x + 3) = 8x\end{align*}2(x+3)=8x

21. \begin{align*}3(x + 5) = −2x\end{align*}3(x+5)=2x

22. \begin{align*}9y = 4(y− 5)\end{align*}9y=4(y5)

Review (Answers)

To see the Review answers, open this PDF file and look for section 3.8.  

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Vocabulary

TermDefinition
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, a(b + c) = ab + ac.
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.

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