44 Chapter 1 Linear Equations and Inequalities in One Variable Section 1.1 Linear Equations in One Variable 1. Linear Equations in One Variable An equation is a statement that indicates that two quantities are equal.The following are equations. x 4 p 3 11 2z 20 All equations have an equal sign. Furthermore, notice that the equal sign separates the equation into two parts, the left-hand side and the right-hand side.A solution to an equation is a value of the variable that makes the equation a true statement. Substituting a solution to an equation for the variable makes the right-hand side equal to the left-hand side. Equation Solution Check 8 Substitute for p. ✓ Right-hand side equals left-hand side. 10 Substitute for z. ✓ Right-hand side equals left-hand side. p 3 11 8 p 3 11 8 3 11 2z 20 10 2z 20 21102 20 The solution set to an equation is the set of all solutions to an equation.We write the solution set using set braces. For example: Equation Solution set p 3 11 586 w2 16 54, 46 Throughout this text we will learn to recognize and solve several different types of equations, but in this chapter, we will focus on the 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 3x 5 20 a 3, b 5, c 20 2x 4 6 can be written as 2x (4) 6 a2, b4, c 6 6x 7 5x 11 can be written as x 7 11 a 1, b 7, c 11 Concepts 1. Linear Equations in One Variable 2. Solving Linear Equations 3. Clearing Fractions and Decimals 4. Conditional Equations, Contradictions, and Identities This equation has one solution. This equation has two solutions.
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