pre-algebra

Pre-Algebra

11-1 Graphing Linear Equations

Learn to identify and graph linear equations.

Pre-Algebra


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.

Pre-Algebra


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

Pre-Algebra


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.

3

3

3

2

2

2

Pre-Algebra


Graph the equation and tell whether it is linear.

A. y = 3x – 1

Example 1A: Graphing Equations

x 3x – 1 y (x, y)

–2

–1

0

1

2

–73(–2) – 13(–1) – 1

3(0) – 13(1) – 1

3(2) – 1

–4

–1

2

5

(–2, –7)

(–1, –4)(0, –1)

(1, 2)(2, 5)

Pre-Algebra


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.

Pre-Algebra



B. y = x3

Example 1B: Graphing Equations

x x3 y (x, y)

–2

–1

0

1

2

–8(–2)3

(–1)3

(0)3

(1)3

(2)3

–1

0

1

8

(–2, –8)

(–1, –1)(0, 0)

(1, 1)(2, 8)

Pre-Algebra


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 0 1 2

y –8 –1 0 1 8

+7 +1 +1 +7

Pre-Algebra


Example 1C: Graphing EquationsGraph the equation and tell whether it is linear.

C. y = – 3x4

Pre-Algebra


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.

3x4

34

Pre-Algebra



D. y = 2

Example 1D: Graphing Equations

For any value of x, y = 2.

x 2 y (x, y)

–2

–1

0

1

2

222

2

2

2

2

2

2

2

(–2, 2)

(–1, 2)(0, 2)

(1, 2)(2, 2)

Pre-Algebra


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.

Pre-Algebra


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.

Pre-Algebra


Example 2 Continued

Pre-Algebra


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.

Example 2 Continued

Pre-Algebra


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.

Pre-Algebra


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

Pre-Algebra


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.

3

3

3

2

2

2

Pre-Algebra


Try This: Example 1CGraph the equation and tell whether it is linear.

C. y = x

x y (x, y)

–8

–6

0

4

8

–8

–6

0

4

8

(–8, –8)

(–6, –6)(0, 0)

(4, 4)(8, 8)

Pre-Algebra


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.

Pre-Algebra


Try This: Example 1D

For any value of x, y = 7.


D. y = 7

x 7 y (x, y)

–8

–4

0

4

8

777

7

7

7

7

7

7

7

(–8, 7)

(–4, 7)(0, 7)

(4, 7)(8, 7)

Pre-Algebra


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.

Pre-Algebra



A. y = 2x + 1

Try This: Example 1A

x 2x + 1 y (x, y)

–2

–1

0

1

2

–32(–2) + 12(–1) + 1

2(0) + 12(1) + 1

2(2) + 1

–1

1

3

5

(–2, –3)

(–1, –1)(0, 1)

(1, 3)(2, 5)

Pre-Algebra


Try This: Example 1A Continued

The equation y = 2x + 1is linear equation because it is the graph of a straight line and each time x increase by 1 unit, y increases by 2 units.

Pre-Algebra


Graphing the equation and tell whether it is linear.

B. y = x2

Try This: Example 1B

x x2 y (x, y)

–2

–1

0

1

2

4(–2)2

1

0

1

4

(–2, 4)

(–1, 1)(0, 0)

(1, 1)(2, 4)

(–1)2

(0)2

(1)2

(2)2

Pre-Algebra


Try This: Example 1B Continued

The equation y = x2 is not a linear equation because its graph is not a straight line.

Pre-Algebra


Try This: Example 2In 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

Pre-Algebra


Try This: Example 2 Continued

t 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.

Pre-Algebra


Try This: Example 2 Continued

x

y

This is a linear equation because when t increases by 1 unit, D increases by 1250 units.

1250

2500

1 2

3750

5000

3 4Time (min)

Dis

tan

ce (

ft)

pre-algebra

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