Solve Equations with Variables on Both Sides

Solve Equations

with Variables

on Both Sides

When you see an equation with variables on both sides you should:

• Move all the terms with variables to one side of the equation by making the term zero on one side.

• Remember that terms with variables can become zero the same way constants become zero: by adding the opposite or using the inverse operation.

• The equation stays balanced when you do the same thing to both sides.

It looks like this:

 $\ 2y\ +\ 27\ =\ \ 5y$  

 $-2y\ \ \ \ \ \ \ \ \ \ \ \ -2y$  

________________________________
 $\ \ \ \ \ \ \ \ \ \ 27\ =\ \ 3y$  

Now we have a one-step equation.

It could look like this:

 $\ \ 9k\ +\ 6\ =\ \ 8k\ +\ 13$  

 $-8k\ \ \ \ \ \ \ \ \ -8k$  

____________________________
  $\ k\ +\ 6\ =\ 13$ 

Now we have a two-step equation.  

Multiple Choice

Let's try it! Choose the correct first step for this equation.

 $\ 7x\ =\ 2x\ -\ 20$  

• Subtract 2x from both sides.

• Subtract 7x from both sides.

• Sutbract 20 from 2x.

Is this what you said?

 $\ 7x\ =\ \ \ 2x\ -\ 20$  
 $-2x\ \ \ \ -2x$  
___________________________
 $\ 5x\ \ =\ -20$  

Multiple Choice

Using the work we just did, find the solution for

 $\ 7x\ =\ 2x\ -\ 20$  

• x = 4

• x = -4

• x = 15

• x = -25

Multiple Choice

Try another one. Choose a correct first step for this equation.

 $14\ +\ 3n\ =\ 5n\ -\ 6$  

• Subtract 5n from both sides.

• Add 14 to 3n

• Subtract 6 from 5n

Is this what you thought?

 $\ 14\ +\ 3n\ =\ \ 5n\ -\ 6$  
 $\ \ \ \ \ \ \ \ -5n\ \ \ \ \ -5n$  
_____________________________
 $\ 14\ -\ 2n\ =\ -6$  

Here's another option:

 $\ 14\ +\ \ 3n\ =\ \ 5n\ -\ 6$  
 $\ \ \ \ \ \ \ \ \ -3n\ \ \ \ \ -3n$  
________________________________
 $\ \ \ \ \ \ \ \ \ \ \ \ 14\ =\ \ 2n\ -\ 6$  
We can move all the variable terms to either side of the equation.

Multiple Choice

Using the math we just saw, solve this equation.

 $\ 14\ +\ 3n\ =\ 5n\ -\ 6$  

• n = 10

• n = -10

• n = -18

• n = -22

Multiple Choice

Let's practice some more. What is the solution to this equation?

 $18\ -\ 2g\ =\ 4g$  

• 3 = g

• g = 12

• 9 = g

• g = 16

Multiple Choice

Try one more:

 $3x\ +\ 5\ =\ 7x\ +\ 4$  

•  $x\ =\ 4$  

•  $x\ =\ -4$  

•  $x\ =\ \frac{1}{4}$  

•  $x\ =\ -\frac{1}{4}$  

Open Ended

On a scale of 1 - 5, with 5 meaning that you understand completely, how do you feel about solving equations with variables on both sides after this lesson? Explain your rating.