Form 1 Unit 9 Lesson 2- Solving Equations with the Variable on Both Sides

Objective

At the end of the lesson, students should be able to solve equations with the variable on both sides.

Sometimes, the unknown quantity will appear on both sides of an equation.
The first step in solving an equation with a variable on both sides is to get the variable on one side. This is done by reversing the addition or subtraction of one of the terms with the variable. In other words, we must add to both sides or subtract from both sides one of the quantities that contains the variable. It is generally easier to add or to subtract the smaller quantity from the larger quantity, so that we work with positive coefficients. Once the variable is on one side only, we can proceed using inverse operations.

Example 1

Solve for x:

a) 4x=2x+16             b) 3x + 2x = 12 – x              c) 5x = 6 + 2x

Solution

b) 3x + 2x = 12 – x

3x + 2x = 12 – x        Simplify.   

5x = 12 – x                 Add x left sides to get the variable on one side.

5x + x  = 12                Simplify.

6x = 12                       Divide by 6 on both sides.

x = 2

c)5x = 6 + 2x                        Subtract -2x from both sides.
5x – 2x = 6 + 2x – 2x           Simplify.
3x = 6                                  Divide by 6 on both sides.

x=2

Check: 3(2) + 2(2) = 12 – 2

                         6+4=10 

                          10=10

Example 2

Solve for the unknown in the following:

a) 5y– 3 = 3y+ 5      b) 12k+15 = 35+2k      c) 2x–6 = 5x+18

Solution

a)5y – 3 – 3y = 3y + 5 – 3y      Simplify.

2y – 3 = 5                                 Add 3 to both one side

2y – 3 + 3 = 5 + 3

2y = 8                                    Divide by 2 on both sides.

y= 4

Check: 5(4) – 3 = 3(4) + 5 

      20-3 = 12+5

                   17 = 17

b) 12k+15=35+2k

Check: 2(–8) – 6 = 5(–8) + 18

            –16 – 6 = –40 + 18

                   –22 =–22

 Example 3

Solve the equation: 2x + 3 = 6 − (2x − 3).

Solution

2x + 3 = 6 − (2x − 3)

Expand to remove the brackets on the right side.

2x + 3 = 6 − 2x + 3              group like terms

2x+2x =6 +3  -3                   Simplify.

4x  =  6                                    Divide both sides by 4.

4x/4 = 6/4

x=1.5

Check: 2(1.5) +3=6− (2(1.5)− 3)  

                         3+3=6 −(3−3)

                          6=6−0

                           6=6