Equations with Variables on Both Sides

Solving Equations with Variables on Both Sides

In this lesson,

Students will be able to:

• Solve linear equations that have variables on both sides.

• Identify special solutions of linear equations.

When solving equations with variables on both sides,

•  The goal is to get the variable on one side (isolate the variable) and the constants on the other sides.

• ** TIP * Always move the smallest coefficient first.

On your desk, solve the following question and I will be coming to check your work.

4x - 7 = -3x

​

​Desktop Math

Multiple Choice

Identify the correct first step in solving this question:

-3(w + 4) = 4w - 5

• Add 5 to both sides

• Divide both sides by -3.

• Distribute -3 into the parentheses

• Subtract 4w from both sides

Solve the following question

On our desk top, solve the following question:

​-3(w + 4) = 4w - 5

Solve the following question

​-3(w + 4) = 4w - 5
-3w - 12 = 4w -5
+ 5 + 5
-3w - 7 = 4w
+ 3w + 3w
-7 = 7w
/-7 /-7
-1 = w

Multiple Choice

Identify the best possible first step to solve this equation:

g - 10 + 7g = 15 + 3g

• Combine like terms

• Subtract 10 from both sides

• Subtract g and 7g from each other

• Divide both sides by 3

Multiple Choice

Solve the following equation for g:

g - 10 + 7g = 15 + 3g

• g = 10

• g = 5

• g = -5

• g = 11/5

Equations with Special Solutions

​Equations do not ALWAYS have just one solution. 

One Solution – The coefficient on sides of the equal sign is different.

 No Solution – Final statement is NOT TRUE.

Many Solutions (Identity) or Infinite Solutions – Final statement is TRUE.

​Equations with 1 solution

7y + 13 = 5y - 3
- 5y - 5y
2y + 13 = -3
-13 -13
2y = -16
/2 /2
y = -8

One Solution

Fill in the Blanks

Solve this equation on our desk and input your solution here:

6n + 1 = 2n - 7

​Equations with no solution

8 + 9p = 9p - 7
- 9p - 9p
8 = -7

This is NOT a true statement, therefore, there is NO Solution!

Open Ended

Solve this question on your desk and identify if it has 1 solution or no solution.

5t + 7 = 5t - 9

​Equations with infinitely many solutions

3(7r - 2) = 21r - 6
21r - 6 = 21r - 6
-21r -21r
-6 = -6

This IS a true statement, therefore, there is infinitely Many Solutions!

Multiple Select

Solve the following equation and determine if it is has 1 solution, no solution or infinitely many solutions:

2(2x - 2) = 4(x - 1)

• NO solution!

• x = 2, One Solution

• -4 = -4

  Infinitely Many Solutions

• 4x = 4x

  Infinitely Many Solutions

Multiple Choice

Solve the following equation for the unknown variable:

5t + 7 = 2t - 9

• No solution

• t = -8

• t = -4

• t = 5

Fill in the Blanks

Identify if the following equation will have 1 solution, no solution or infinitely Many solution:

6w + 3 - 10w = 7w - 8

Open Ended

When solving equations with variables on both sides, what is the ultimate goal for solving the equation?

Fill in the Blanks

Solve the following equation and determine if it has 1 solution, no solution or infinitely many solutions:

2(3x + 6) = 3(2x - 6)

Multiple Choice

Solve the following equation:

8(3g + 2) - 3g = 3(5g - 4) - 2

• g = -5

• g = -11

• g = -6

• g = 6