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

​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

Open Ended

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

Multiple Choice

Solve the following equation:

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

• g = -5

• g = -11

• g = -6

• g = 6