Graphing Linear Equations 11-1 Graphing Linear Equations Warm Up Problem of the Day Lesson Presentation Pre-Algebra

Graphing Linear Equations Pre-Algebra 11-1 Graphing Linear Equations Warm Up Solve each equation for y. 1. 6y – 12x = 24 2. –2y – 4x = 20 3. 2y – 5x = 16 4. 3y + 6x = 18 y = 2x + 4 y = –2x – 10 y = x + 8 5 2 y = –2x + 6

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

Learn to identify and graph linear equations.

Vocabulary linear equation

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 (x1, y1) and (x2, y2), choose an x-value between x1 and x2 and find the corresponding y-value.

Reading Math Read x1 as “x sub one” or “x one.”

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 3 2 3 2 3

Additional Example 1A: Graphing Equations Graph the equation and tell whether it is linear. A. y = 3x – 1 x 3x – 1 y (x, y) –2 –1 1 2 3(–2) – 1 –7 (–2, –7) 3(–1) – 1 –4 (–1, –4) 3(0) – 1 –1 (0, –1) 3(1) – 1 2 (1, 2) 3(2) – 1 5 (2, 5)

Additional Example 1A Continued The equation y = 3x – 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.

Additional Example 1B: Graphing Equations Graph the equation and tell whether it is linear. B. y = x3 x x3 y (x, y) –2 –1 1 2 (–2)3 –8 (–2, –8) (–1)3 –1 (–1, –1) (0)3 (0, 0) (1)3 1 (1, 1) (2)3 8 (2, 8)

Additional Example 1B Continued The equation y = x3 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 –2 –1 1 2 y –8 8 +7 +1 +1 +7

Additional Example 1C: Graphing Equations Graph the equation and tell whether it is linear. C. y = – 3x 4

Additional Example 1 Continued The equation y = – 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 or y decreases by 3 each time x increases by 4. 3x 4 3

Additional Example 1D: Graphing Equations Graph the equation and tell whether it is linear. D. y = 2 x 2 y (x, y) –2 –1 1 2 2 (–2, 2) 2 2 (–1, 2) 2 2 (0, 2) 2 2 (1, 2) 2 2 (2, 2) For any value of x, y = 2.

Additional Example 1D 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.

Try This: Example 1A Graph the equation and tell whether it is linear. A. y = 2x + 1 x 2x + 1 y (x, y) –2 –1 1 2 2(–2) + 1 –3 (–2, –3) 2(–1) + 1 –1 (–1, –1) 2(0) + 1 1 (0, 1) 2(1) + 1 3 (1, 3) 2(2) + 1 5 (2, 5)

Try This: Example 1A Continued The equation y = 2x + 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.

Try This: Example 1B Graphing the equation and tell whether it is linear. B. y = x2 x x2 y (x, y) –2 –1 1 2 (–2)2 4 (–2, 4) (–1)2 1 (–1, 1) (0)2 (0, 0) (1)2 1 (1, 1) (2)2 4 (2, 4)

Try This: Example 1B Continued The equation y = x2 is not a linear equation because its graph is not a straight line.

Try This: Example 1C Graph the equation and tell whether it is linear. C. y = x x y (x, y) –8 –6 4 8 –8 (–8, –8) –6 (–6, –6) (0, 0) 4 (4, 4) 8 (8, 8)

Try This: Example 1C 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.

Try This: Example 1D Graph the equation and tell whether it is linear. D. y = 7 x 7 y (x, y) –8 –4 4 8 7 7 (–8, 7) 7 7 (–4, 7) 7 7 (0, 7) 7 7 (4, 7) 7 7 (8, 7) For any value of x, y = 7.

Try This: Example 1D 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.

Additional Example 2: Sports Application A lift on a ski slope rises according to the equation a = 130t + 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.

Additional Example 2 Continued

Additional Example 2 Continued

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.

Try This: Example 2 In an amusement park ride, a car travels according to the equation D = 1250t 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

Try This: Example 2 Continued D =1250t 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.

Try This: Example 2 Continued 5000 3750 Distance (ft) 2500 1250 x 1 2 3 4 Time (min) This is a linear equation because when t increases by 1 unit, D increases by 1250 units.

Lesson Quiz Graph each equation and tell whether it is linear. 1. y = 3x – 1 2. y = x 3. y = x2 – 3 yes 14 yes no