The Addition/Subtraction Principle of Equality 4.2 The Addition/Subtraction Principle of Equality Determine whether a given equation is linear. Solve linear equations in one variable using the addition/subtraction principles of equality. Solve equations with variables on both sides of the equal sign. Solve application problems.

Objective 1 Determine whether a given equation is linear.

Definition Linear equation: An equation in which each variable term is a monomial of degree 1. This means that linear equations contain constants and variables terms that have a single variable raised to an exponent of 1. All other equations are called nonlinear equations.

Example 1 Determine whether the given equation is linear or nonlinear. a. 5x + 9 = 24 b. 2x – 9x2 = 15x + 7

Linear equation in one variable: Nonlinear equation in one variable: Linear equation in two variables: Nonlinear equation in two variables:

Objective 2 Solve linear equations in one variable using the addition/subtraction principle of equality.

the equal sign acting as the pivot point. Let’s use a technique for solving more complex equations called the balance technique. We can illustrate the nature of the technique by imagining an equation as a balanced scale with the equal sign acting as the pivot point. x + 7 9 =

When weight is added or removed on one side of the scale, it tips out of balance. In the figure, we add 4 to the left side of the scale, tipping it out of balance. x + 7 + 4 9 ≠ To maintain balance, the same weight must be added or removed on both sides of the scale. Adding 4 to the right side of our scale balances it. x + 7 + 4 9 +4 =

Rule The Addition/Subtraction Principle of Equality. The same amount can be added to or subtracted from both sides of an equation without affecting its solution(s).

The addition principle tells us that adding the same number to both sides will not change the fact that 2 is the solution. So, when we add 4 to both sides, which balances the equation, 2 should still be the solution, and it is.

In the case of x + 7 = 9, we use the addition principle to rewrite the equation as x = 2. Notice that adding – 7 to both sides (or subtracting 7 from both sides) achieves this goal.

Procedure To use the addition/subtraction principle of equality to clear a term, add the additive inverse of the term to both sides of the equation (or subtract the term from both sides of the equation).

Example 2 Solve and check. a. x – 8 = – 15 b. 14 = m + 5

Example 3 Solve and check. 5n – 14 – 4n = – 12 + 7

Example 4 Solve and check. 3(y – 5) – 2y = – 20 + 2

Objective 3 Solve equations with variables on both sides of the equal sign.

Example 5 Solve and check. 7x – 9 = 6x – 11

Example 6 Solve and check. y – (2y + 9) = 6(y – 2) – 8y

Procedure To solve a linear equation in one variable: Simplify both sides of the equation as needed. Distribute to clear parentheses. Combine like terms. 2. Use the addition/subtraction principle of equality so that all variable terms are on one side of the equation and all constants are on the other side. Then combine like terms. Note: Clearing the variable term with the lesser coefficient will avoid negative coefficients.

Objective 4 Solve application problems.

Example 7 Laura wants to buy a new car stereo that costs $275. She currently has $142. How much more does she need?