Modeling Equations with Variables on Both Sides

Solve Equations with Variables on Both Sides

Lesson 7.1

Open Ended

What are the things you wish 2021 would bring to your life?

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.