Section 2.1 Addition, Subtraction, Multiplication, and Division Properties of Equality 121 2a2 4 0 Answers 3. Yes 4. No 5. Yes 6. No In the study of algebra, you will encounter a variety of equations. In this chapter, we will focus on a specific type of equation called a linear equation in one variable. Definition of a Linear Equation in One Variable Let a, b, and c be real numbers such that a 0. A linear equation in one variable is an equation that can be written in the form ax b c Note: A linear equation in one variable is often called a first-degree equation because the variable x has an implied exponent of 1. Examples Notes 2x 4 20 a 2, b 4, c 20 3x 5 16 can be written as 3x (5) 16 a 3, b 5, c 16 5x 9 4x 1 can be written as x 9 1 a 1, b 9, c 1 2. Addition and Subtraction Properties of Equality If two equations have the same solution then the equations are equivalent. For example, the following equations are equivalent because the solution for each equation is 6. Equivalent Equations Check the Solution 6 ✔ ✔ ✔ 2x 5 7 2162 5 7 1 12 5 7 2x 12 2162 12 1 12 12 x 6 6 6 1 6 6 To solve a linear equation, the goal is to find all values of x that make the equation true. One general strategy for solving an equation is to rewrite it as an equivalent but simpler equation. This process is repeated until the equation can be written in the form x number. The addition and subtraction properties of equality help us do this. Addition and Subtraction Properties of Equality Let a, b, and c represent algebraic expressions. 1. Addition property of equality: If then 2. *Subtraction property of equality: If then a b, a c b c a b, *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 a c b c ax b c, Concept Connections State whether the equation is a linear equation in one variable. 3. 5y 1 0 4. 5. 6 3x 0 6. 6t 8y 0
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