142 Chapter 3 Solving Equations Definition of a Linear Equation in One Variable Let a, b, and c be 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 a3, b5, 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 Given the equation x 3, we can easily determine that the solution is 3. The solution to the equation 2x 14 20 is also 3. These two equations are called equivalent equations because they have the same solution. However, while the solution to x 3 is obvious, the solution to 2x 14 20 is not obvious. Our goal in this chapter is to learn how to solve equations. To solve an equation we use algebraic principles to write an equation like 2x 14 20 in an equivalent but simpler form, such as x 3. The addition and subtraction properties of equality are the first tools we will use to solve an equation. Addition and Subtraction Properties of Equality Let a, b, and c represent algebraic expressions. 1. The addition property of equality: If a b, then, a c b c 2. The subtraction property of equality: If a b, then, a c b c The addition and subtraction properties of equality indicate that adding or subtracting the same quantity to each side of an equation results in an equivalent equation. This is true because if two equal quantities are increased (or decreased) by the same amount, then the resulting quantities will also be equal (Figure 3-1). Figure 3-1 50 lb 50 lb 20 lb 50 lb 20 lb 50 lb
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