12 1 Graphing Linear Equations Warm Up Problem
12 -1 Graphing Linear Equations Warm Up Problem of the Day Lesson Presentation Course 3
12 -1 Graphing Linear Equations Warm Up Solve each equation for y. 1. 6 y – 12 x = 24 y = 2 x + 4 2. – 2 y – 4 x = 20 y = – 2 x – 10 3. 2 y – 5 x = 16 y = 5 x + 8 2 4. 3 y + 6 x = 18 y = – 2 x + 6 Course 3
12 -1 Graphing Linear Equations Problem of the Day The same photo book of Niagara Falls costs $5. 95 in the United States and $8. 25 in Canada. If the exchange rate is $1. 49 in Canadian dollars for each U. S. dollar, in which country is the book a better deal? Canada Course 3
12 -1 Graphing Linear Equations Learn to identify and graph linear equations. Course 3
12 -1 Graphing Insert Lesson Title Here Linear Equations Vocabulary linear equation Course 3
12 -1 Graphing Linear Equations A linear equation is an equation whose solutions fall on a line on the coordinate plane. All solutions of a particular linear equation fall on the line, and all the points on the line are solutions of the equation. To find a solution that lies between two points (x 1, y 1) and (x 2, y 2), choose an x-value between x 1 and x 2 and find the corresponding y-value. Course 3
12 -1 Graphing Insert Lesson Title Here Linear Equations Reading Math Read x 1 as “x sub one” or “x one. ” Course 3
12 -1 Graphing Linear Equations If an equation is linear, a constant change in the x-value corresponds to a constant change in the y-value. The graph shows an example where each time the x-value increases by 3, the y-value increases by 2. 2 23 23 3 Course 3
12 -1 Graphing Linear Equations Additional Example 1 A: Graphing Equations Graph the equation and tell whether it is linear. y = 3 x – 1 x – 2 – 1 0 1 2 Course 3 3 x – 1 y (x, y) 3(– 2) – 1 3(– 1) – 1 – 7 – 4 – 1 2 5 (– 2, – 7) (– 1, – 4) (0, – 1) 3(0) – 1 3(1) – 1 3(2) – 1 (1, 2) (2, 5)
12 -1 Graphing Linear Equations Additional Example 1 A Continued The equation y = 3 x – 1 is a linear equation because it is the graph of a straight line and each time x increases by 1 unit, y increases by 3 units. Course 3
12 -1 Graphing Insert Lesson Title Here Linear Equations Caution! Be careful when graphing each ordered pair. Double check each point you plot. Course 3
12 -1 Graphing Linear Equations Additional Example 1 B: Graphing Equations Graph the equation and tell whether it is linear. y = x 3 x – 2 – 1 0 1 2 Course 3 x 3 y (x, y) (– 2)3 (– 1)3 (0)3 (1)3 (2)3 – 8 – 1 0 1 8 (– 2, – 8) (– 1, – 1) (0, 0) (1, 1) (2, 8)
12 -1 Graphing Linear Equations Additional Example 1 B Continued The equation y = x 3 is not a linear equation because its graph is not a straight line. Also notice that as x increases by a constant of 1 unit, the change in y is not constant. x y – 2 – 8 – 1 +7 Course 3 +1 0 0 1 1 +1 +7 2 8
12 -1 Graphing Linear Equations Additional Example 1 C: Graphing Equations Graph the equation and tell whether it is linear. y=– Course 3 3 x 4
12 -1 Graphing Linear Equations Additional Example 1 Continued The equation y = – 3 x 4 is a linear equation because the points form a straight line. Each time the value of x increases by 1, the value of y decreases by 3 4 or y decreases by 3 each time x increases by 4. Course 3
12 -1 Graphing Linear Equations Additional Example 1 D: Graphing Equations Graph the equation and tell whether it is linear. y=2 x – 2 – 1 0 1 2 2 y (x, y) 2 2 2 2 (– 2, 2) (– 1, 2) (0, 2) 2 2 2 For any value of x, y = 2. Course 3 (1, 2) (2, 2)
12 -1 Graphing Linear Equations Additional Example 1 D Continued The equation y = 2 is a linear equation because the points form a straight line. As the value of x increases, the value of y has a constant change of 0. Course 3
12 -1 Graphing Linear Equations Check it Out: Example 1 A Graph the equation and tell whether it is linear. y = 2 x + 1 x – 4 – 2 0 2 4 Course 3 2 x + 1 y (x, y) 2(– 4) + 1 2(– 2) + 1 – 7 – 3 1 5 9 (– 4, – 7) (– 2, – 3) (0, 1) 2(0) + 1 2(2) + 1 2(4) + 1 (2, 5) (4, 9)
12 -1 Graphing Linear Equations Check It Out: Example 1 A Continued The equation y = 2 x + 1 is linear equation because it is the graph of a straight line and each time x increase by 1 unit, y increases by 2 units. Course 3
12 -1 Graphing Linear Equations Check It Out: Example 1 B Graphing the equation and tell whether it is linear. y = x 2 x – 2 – 1 0 1 2 Course 3 x 2 (– 2)2 (– 1)2 (0)2 (1)2 (2)2 y (x, y) 4 1 0 1 4 (– 2, 4) (– 1, 1) (0, 0) (1, 1) (2, 4)
12 -1 Graphing Linear Equations Check It Out: Example 1 B Continued The equation y = x 2 is not a linear equation because its graph is not a straight line. Course 3
12 -1 Graphing Linear Equations Check It Out: Example 1 C Graph the equation and tell whether it is linear. y=x x – 8 – 6 0 4 8 Course 3 y (x, y) – 8 – 6 0 4 8 (– 8, – 8) (– 6, – 6) (0, 0) (4, 4) (8, 8)
12 -1 Graphing Linear Equations Check It Out: Example 1 C Continued The equation y = x is a linear equation because the points form a straight line. Each time the value of x increases by 1, the value of y increases by 1. Course 3
12 -1 Graphing Linear Equations Check It Out: Example 1 D Graph the equation and tell whether it is linear. D. y = 7 x – 8 – 4 0 4 8 7 y (x, y) 7 7 7 7 (– 8, 7) (– 4, 7) (0, 7) 7 7 7 For any value of x, y = 7. Course 3 (4, 7) (8, 7)
12 -1 Graphing Linear Equations Check It Out: Example 1 D Continued The equation y = 7 is a linear equation because the points form a straight line. As the value of x increases, the value of y has a constant change of 0. Course 3
12 -1 Graphing Linear Equations Additional Example 2: Sports Application A lift on a ski slope rises according to the equation a = 130 t + 6250, where a is the altitude in feet and t is the number of minutes that a skier has been on the lift. Five friends are on the lift. What is the altitude of each person if they have been on the ski lift for the times listed in the table? Draw a graph that represents the relationship between the time on the lift and the altitude. Course 3
12 -1 Graphing Linear Equations Additional Example 2 Continued Course 3
12 -1 Graphing Linear Equations Additional Example 2 Continued Course 3
12 -1 Graphing Linear Equations Additional Example 2 Continued The altitudes are: Anna, 6770 feet; Tracy, 6640 feet; Kwani, 6510 feet; Tony, 6445 feet; George, 6380 feet. This is a linear equation because when t increases by 1 unit, a increases by 130 units. Note that a skier with 0 time on the lift implies that the bottom of the lift is at an altitude of 6250 feet. Course 3
12 -1 Graphing Linear Equations Check It Out: Example 2 In an amusement park ride, a car travels according to the equation D = 1250 t where t is time in minutes and D is the distance in feet the car travels. Below is a chart of the time that three people have been in the cars. Graph the relationship between time and distance. How far has each person traveled? Rider Time Ryan 1 min Greg 2 min Colette 3 min Course 3
12 -1 Graphing Linear Equations Check It Out: Example 2 Continued t D =1250 t D (t, D) 1 1250(1) 1250 (1, 1250) 2 1250(2) 2500 (2, 2500) 3 1250(3) 3750 (3, 3750) The distances are: Ryan, 1250 ft; Greg, 2500 ft; and Collette, 3750 ft. Course 3
12 -1 Graphing Linear Equations Check It Out: Example 2 Continued y Distance (ft) 5000 3750 2500 1250 x 1 2 3 Time (min) 4 This is a linear equation because when t increases by 1 unit, D increases by 1250 units. Course 3
12 -1 Graphing Insert Lesson Linear Title Equations Here Lesson Quiz Graph each equation and tell whether it is linear. 1. y = 3 x – 1 yes 1 x yes 4 3. y = x 2 – 3 no 2. y = Course 3
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