Section 1.1 Linear Equations in One Variable 45 2. Solving Linear Equations To solve a linear equation, the goal is to simplify the equation to isolate the variable. Each step used in simplifying an equation results in an equivalent equation. Equivalent equations have the same solution set. For example, the equations and are equivalent because is the solution set for both equations. To solve an equation, we may use the addition, subtraction, multiplication, and division properties of equality. These properties state that adding, subtracting, multiplying, or dividing the same quantity on each side of an equation results in an equivalent equation. Addition and Subtraction Properties of Equality Let a, b, and c represent real numbers. Addition property of equality: If then *Subtraction property of equality: If a b, then a c b c. a b, a c b c. 2x 4 526 2x 3 7 *The subtraction property of equality follows directly from the addition property, because subtraction is defined in terms of addition. a 1c2 b 1c2 If then, a c b c Multiplication and Division Properties of Equality Let a, b, and c represent real numbers with c 0. Multiplication property of equality: If then a b, a c b c. *Division property of equality: If then a c b c a b, . *The division property of equality follows directly from the multiplication property, because division is defined as multiplication by the reciprocal. If then, a c b c a 1 c b 1 c Solving a Linear Equation Example 1 Solve the equation. Solution: 12 x 40 To isolate x, subtract 12 from both sides. 12 x 40 12 12 x 40 12 x 28 Simplify.
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