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3p + 4q = 9 3(3) + 4q = 9 9 + 4q = 9 4q = 0 q=0 Answer: p = 3, q = 0 Substitution method Sometimes a system is more easily solved by the substitution method. This method involves substituting one equation into another. Example 5: Solve for x and y. x=y+8 x + 3y = 48 From the first equation, substitute (y + 8) for x in the second equation. (y + 8) + 3y = 48 Now solve for y. Simplify by combining y’s. 4y + 8 = 48 -8 -8 4y = 40 4y 40 = 4 4 y = 10 Now insert y = 10 in one of the original equations.

Working Equations An equation is a mathematical sentence, a relationship between numbers and/or symbols. Axioms of equality For all real numbers a, b, and c, the following are some basic rules for using the equal sign. ■ Reflexive axiom: a = a. Therefore, 4 = 4. ■ Symmetric axiom: If a = b, then b = a. Therefore, if 2 + 3 = 5, then 5 = 2 + 3. ■ Transitive axiom: If a = b and b = c, then a = c. Therefore, if 1 + 3 = 4 and 4 = 2 + 2, then 1 + 3 = 2 + 2. Chapter 4: Equations, Ratios, and Proportions 41 ■ Additive axiom: If a = b and c = d, then a + c = b + d.

3 × 10 )(2 × 10 ) = 4 8 5 4. 5 5. 12 or 2 2 6. 125 7. 8 8. 25% Answers: 1. –3 2. 36 3. 12 9 12 5 5 12 9. 6 × 10 Chapter 3 TERMINOLOGY, SETS, AND EXPRESSIONS Chapter Check-In ❑ Set theory ❑ Algebraic expressions ❑ Evaluating expressions nderstanding the language of algebra and how to work with algebraic expressions will give you a good foundation for learning the rules of U algebra. Some Basic Language The two sections that follow will give you a “heads up” on some ways of writing the same operation and the common algebraic use of some letters.