Section 2.1 Solving Equations: The Addition Principle 1 Equations and Solutions ESSENTIALS An equation is a number sentence that says that the expressions on either side of the equals sign, =, represent the same number. Any replacement for the variable that makes an equation true is called a solution of the equation. To solve an equation means to find all of its solutions. Examples 2 5 7 The equation is true. 9 3 3 The equation is false. x 8 11 The equation is neither true nor false, because we do not know what number x represents. GUIDED LEARNING EXAMPLE 1 YOUR TURN 1 Determine whether the equation is true, 4 6 2 The equation is false. Determine whether the equation is true, 5 9 4 EXAMPLE 2 YOUR TURN 2 Determine whether the equation is true, Determine whether the equation is true, 13 7 5 15 12 4 7 7 The equation is true. EXAMPLE 3 YOUR TURN 3 Determine whether the equation is true, Determine whether the equation is true, x 5 14 73 x The equation is neither true nor false, because we do not know what number x represents.
2 Section 2.1 Solving Equations: The Addition Principle EXAMPLE 4 YOUR TURN 4 Determine whether 6 is a solution of 10 y 16. Determine whether 22 is a solution 10 y 16 Writing the equation of x 2 20. 10 16 Substituting 6 for TRUE The statement 16 16 is. true / false 6 a solution of 10 y 16. is / is not EXAMPLE 5 YOUR TURN 5 Determine whether 13 is a solution of 9a 107. 9a 107 Writing the equation 9 107 Substituting 13 for FALSE The statement 117 107 is. true / false 13 a solution of 9a 107. is / is not a y Determine whether 12 is a solution of 7x 84. YOUR NOTES Write your questions and additional notes. Answers: 1. True 2. False 3. Neither 4. No 5. Yes
Section 2.1 Solving Equations: The Addition Principle 3 Using the Addition Principle ESSENTIALS Equations with the same solutions are called equivalent equations. The Addition Principle for Equations: For any real numbers a, b, and c, a b is equivalent to a c b c. Examples Solve: w 9 2. w 92 w 9 9 2 9 w 0 11 w 11 Solve: x 3 7. x 3 7 x 3 3 7 3 x 0 10 x 10 GUIDED LEARNING EXAMPLE 1 YOUR TURN 1 Solve: y 13 2. y 13 2 y 13 2 Subtracting 13 on both sides y 0 15 Simplifying The solution is 15. y Identity property of 0 Solve: x 9 15. EXAMPLE 2 YOUR TURN 2 Solve: b 7 12. Solve: a 3 9. b 7 12 b 7 12 Adding 7 on both sides b 0 19 Simplifying b Identity property of 0 Check: b 7 12 19 7 12 12 The solution is 19. TRUE
4 Section 2.1 Solving Equations: The Addition Principle EXAMPLE 3 YOUR TURN 3 Solve: 5.3 b 2.4. 5.3 b 2.4 5.3 b 2.4 Adding 2.4 on both sides b Check: 5.3 b 2.4 5.3 2.9 2.4 5.3 The solution is. TRUE Solve: 7.6 x 4.2. EXAMPLE 4 YOUR TURN 4 Solve: 1 x 2. 2 5 1 2 x 2 5 1 2 x 2 5 2 1 5 x 5 2 2 4 5 x 10 10 1 Subtracting on both sides 2 Multipying by 1 to obtain a common denominator Solve: 2 w 3. 3 4 x 9 The number checks. 10 The solution is. YOUR NOTES Write your questions and additional notes. Answers: 1. 24 2. 6 3. 3.4 4. 17 12
Section 2.1 Solving Equations: The Addition Principle 5 Practice Exercises Readiness Check Choose from the column on the right the most appropriate first step in solving each equation. 1. 11 x 7 a) Add 21 on both sides. 2. 6 x 21 b) Add 7 on both sides. 3. x 6 10 c) Subtract 10 on both sides. 4. x 7 5 d) Subtract 6 on both sides. e) Add 7 on both sides. f) Add 6 on both sides. Equations and Solutions Determine whether the given number is a solution of the given equation. 5. 14; x 19 43 6. 28; x 19 9 x 7. 5; 6x 30 8. 24; 4 6 x 10. y 9. 11; 8 6 92 4; 6 3 42
6 Section 2.1 Solving Equations: The Addition Principle Using the Addition Principle Solve using the addition principle. Don t forget to check! 11. x 7 9 12. m 9 3 13. r 13 4 14. 10 y 3 15. 1 3 r 16. 2 10 3 5 x 4 6 17. 6.2 4.7 y 18. 2.2 8.9 x Answers: 1. e) 2. d) 3. f) 4. b) 5. No 6. Yes 7. Yes 8. Yes 9. No 10. No 11. 2 12. 12 13. 17 14. 13 15. 4 5 16. 1 17. 10.9 18. 6.7 12