Solving Equations by Addition or Subtraction

Equivalent Equations
An equivalent equation can be obtained from an existing equation in one of four ways.

• Add the same term to both sides of the equation.
• Subtract the same term from both sides.
• Multiply by the same term on both sides.
• Divide by the same term on both sides.

The following four equations are equivalent to x = 5

• Add 3 to both sides: x + 3 = 8
• Subtract 3 from both sides: x – 3 = 2
• Multiply by 3 on both sides: 3x = 15
• Divide by 3 on both sides:

Solving equations

We can use equivalent equations to solve an equation. The solution is obtained when the variable is by itself on one side of the equation. The objective, then, is to use equivalent equations to isolate the variable on one side of the equation.
Consider the equation x + 6 = 14. For it to be considered solved, the x has to be on a side by itself. How can you get rid of the +6 that is also on that side? Remember that a term and its additive inverse add up to 0. The additive inverse of +6 is –6. To write an equivalent equation, subtract 6 from both sides.

Example:

Solve x + 6 = 14

Solution:

x + 6 = 14
x + 6 – 6 = 14 – 6 (Subtract 6 from both sides)
x = 8

Check:
x + 6 = 14
8 + 6 = 14 (Substitute x = 8 into original equation)


To solve the equation y – 4 = 12, we would need to write an equivalent equation with y on a side by itself. To get rid of –4, we would need to add 4 to both sides of the equation.

Example:

Solve y – 4 = 12

Solution:

y – 4 = 12
y – 4 + 4 = 12 + 4 (Add 4 to both sides)
y = 16

Check:

y – 4 = 12
16 – 4 = 12 (Substitute y =16 into original equation)