1. Solving Linear Equations by Graphing (3-2) Objective: Solve equations by graphing. Estimate solutions to an equation by graphing.

2. Solve by Graphing A linear function is a function for which the graph is a line. The simplest linear function is f(x) = x and is called the parent function of the family of linear functions. A family of graphs is a group of graphs with one or more similar characteristics.

3. Linear Function Parent function: f(x) = x Type of graph: line Domain: all real numbers Range: all real numbers

4. Linear Function The solution or root of an equation is any value that makes the equation true. A linear equation has at most one root. You can find the root of an equation by graphing its related function. To write the related function for an equation, replace 0 with f(x). –Linear Equation: 2x – 8 = 0 –Related Function: f(x) = 2x – 8 or y = 2x – 8

5. Linear Function Values of x for which f(x) = 0 are called zeros of the function f. The zero of a function is located at the x-intercepts of the function. The root of an equation is the value of the x-intercept.

6. Linear Function Example:  4 is the x-intercept of 2x – 8 = 0.  4 is the solution of 2x – 8 = 0.  4 is the root of 2x – 8 = 0.  4 is the zero of f(x) = 2x – 8. y = 2x – 8

7. Example 1 Solve each equation algebraically and graphically. a.0 = ½ x + 3 -3 -3 = ½ x2 -6 = x x = -6 Graph y = ½ x + 3 XY 0 3 2 4 45 x-int = -6 x = -6

8. Example 1 Solve each equation algebraically and graphically. b.2 = 1 / 3 x + 3 -3 -1 = 1 / 3 x3 -3 = x x = -3 Graph y = 1 / 3 x + 1 XY 0 1 3 2 63 x-int = -3 x = -3 2 = 1 / 3 x + 3 -2 0 = 1 / 3 x + 1

9. Linear Function For equations with the same variable on each side of the equation, use addition or subtraction to get the terms with variables on one side. Then solve.

10. Example 2 Solve each equation algebraically and graphically. a.2x + 5 = 2x + 3 -2x 5 = 3 No Solution 2x + 5 = 2x + 3 -2x 5 = 3 -3 2 = 0 Graph y = 2 Since there is no x- intercept, there is no solution.

11. Example 2 Solve each equation algebraically and graphically. b.5x – 7 = 5x + 2 -5x -7 = 2 No Solution 5x – 7 = 5x + 2 -5x -7 = 2 -2 -9 = 0 Graph y = -9 Since there is no x- intercept, there is no solution.

12. Estimate Solutions by Graphing Graphing may provide only an estimate. In these cases, solve algebraically to find the exact solution.

13. Example 3 Kendra’s class is selling greeting cards to raise money for new soccer equipment. They paid $115 for the cards, and they are selling each card for $1.75. The function y = 1.75x – 115 represents their profit y for selling x greeting cards. Find the zero of this function. Describe what this value means in this context. XY 80 100 25 60 X-int is around 65? 0 = 1.75x – 115 +115 115 = 1.75x 1.75 65.7 = x The class must sell at least 66 cards to make any profit.

14. Check Your Progress Choose the best answer for the following. A.Solve -2 = 2 / 3 x + 4 algebraically. A.x = -4 B.x = -9 C.x = 4 D.x = 9 -2 = 2 / 3 x + 4 -4 -6 = 2 / 3 x 3 / 2

15. Check Your Progress Choose the best answer for the following. B.Solve 6 = -¾ x + 9 by graphing. A.x = 4; B.x = -4; C.x = -3; D.x = 3; 6 = -¾ x + 9 -6 0 = -¾ x + 3 Graph y = -¾ x + 3 XY 0 3 4 0 -46

16. Check Your Progress Choose the best answer for the following. A.Solve -3x + 6 = 7 – 3x algebraically. A.x = 0 B.x = 1 C.x = -1 D.no solution -3x + 6 = 7 – 3x +3x 6 = 7

17. Check Your Progress Choose the best answer for the following. B.Solve 4 – 6x = -6x + 3 by graphing. A.x = -1; B.x = 1; C.x = 1; D.no solution 4 – 6x = -6x + 3 +6x 4 = 3 -3 1 = 0 Graph y = 1

18. Check Your Progress Choose the best answer for the following. –On a trip to his friend’s house, Raphael’s average speed was 45 miles per hour. The distance that Raphael is from his friend’s house at a certain moment in the trip can be represented by d = 150 – 45t, where d represents the distance in miles and t is the time in hours. Find the zero of this function. A.3; Raphael will arrive at his friend’s house in 3 hours. B.3 1 / 3 ; Raphael will arrive at his friend’s house in 3 hours and 20 minutes. C.3 1 / 3 ; Raphael will arrive at his friend’s house in 3 hours 30 minutes. D.4; Raphael will arrive at his friend’s house in 4 hours. 0 = 150 – 45t -150 -150 = -45t -45