120 Chapter 2 Linear Equations and Inequalities Section 2.1 Addition, Subtraction, Multiplication, 1. Definition of a Linear Equation in One Variable An equation is a statement that indicates that two expressions are equal. The following are equations. x 5 y 2 12 4z 28 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 into an equation for the variable makes the right-hand side equal to the left-hand side. Equation Solution Check 5 Substitute 5 for x. Right-hand side equals ✔ left-hand side. x 5 x 5 5 5 10 Substitute 10 for y. Right-hand side equals ✔ left-hand side. y 2 12 y 2 12 10 2 12 4z 28 4z 28 7 Substitute 7 for z. Right-hand side equals 4172 28 ✔ left-hand side. Answers 1. No 2. Yes Determining Whether a Number Is a Solution to an Equation Example 1 12 Determine whether the given number is a solution to the equation. a. 4x 7 5; b. 4 6w 14; 3 Solution: a. Substitute for x. Simplify. ✔ Right-hand side equals the left-hand side. Thus, is a solution to the equation 4x 7 5. b. Substitute 3 for w. Simplify. Right-hand side does not equal left-hand side. Thus, 3 is not a solution to the equation 4 6w 14. 4x 7 5 2 7 5 4 6w 14 4 6132 14 4 18 14 4 4 12 5 5 12 4112 2 7 5 Classroom Example: p. 129, Exercise 14 and Division Properties of Equality Concepts 1. Definition of a Linear Equation in One Variable 2. Addition and Subtraction Properties of Equality 3. Multiplication and Division Properties of Equality 4. Translations Avoiding Mistakes Be sure to notice the difference between solving an equation versus simplifying an expression. For example, 2x 1 7 is an equation,whose solution is 3,while 2x 1 7 is an expression that simplifies to 2x 8. Skill Practice Determine if the number given is a solution to the equation. 1. 4x 1 7; 3 2. 9 2y 5; 2
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