Equations and Solutions

Transcription

1 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 The equation is true 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, The equation is false. Determine whether the equation is true, EXAMPLE 2 YOUR TURN 2 Determine whether the equation is true, Determine whether the equation is true, The equation is true. EXAMPLE 3 YOUR TURN 3 Determine whether the equation is true, Determine whether the equation is true, x x The equation is neither true nor false, because we do not know what number x represents.

2 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 Substituting 6 for TRUE The statement 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 a 107 Writing the equation Substituting 13 for FALSE The statement 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

3 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 w 0 11 w 11 Solve: x 3 7. x 3 7 x x 0 10 x 10 GUIDED LEARNING EXAMPLE 1 YOUR TURN 1 Solve: y 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 EXAMPLE 2 YOUR TURN 2 Solve: b 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 The solution is 19. TRUE

4 4 Section 2.1 Solving Equations: The Addition Principle EXAMPLE 3 YOUR TURN 3 Solve: 5.3 b b b 2.4 Adding 2.4 on both sides b Check: 5.3 b The solution is. TRUE Solve: 7.6 x 4.2. EXAMPLE 4 YOUR TURN 4 Solve: 1 x x x x x Subtracting on both sides 2 Multipying by 1 to obtain a common denominator Solve: 2 w x 9 The number checks. 10 The solution is. YOUR NOTES Write your questions and additional notes. Answers:

5 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 x 7 a) Add 21 on both sides 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 ; x ; x 19 9 x 7. 5; 6x ; 4 6 x 10. y 9. 11; ;

6 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 m r y r x y x Answers: 1. e) 2. d) 3. f) 4. b) 5. No 6. Yes 7. Yes 8. Yes 9. No 10. No