Chapter 3 Section 2 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.

Chapter Chapter 33Section Section 22

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley


Graphing Linear Equations in Two Variables

11

44

33

22

55

3.23.23.23.2Graph linear equations by plotting ordered pairs.Find intercepts.Graph linear equations of the form Ax + By = 0.Graph linear equations of the form y = k or x = k.Use a linear equation to model data.


Objective 11

Slide 3.2 - 3

Graph linear equations by plotting ordered pairs.


Graph linear equations by plotting ordered pairs.

Infinitely many ordered pairs satisfy a linear equation in two variables. We find these ordered-pair solutions by choosing as many values of x (or y) as we wish and then completing each ordered pair.

Some solutions of the equation x + 2y = 7 are graphed below.

Slide 3.2 - 4


Graph linear equations by plotting ordered pairs. (cont’d)

Notice that the points plotted in the previous graph all appear to lie on a straight line, as shown below.

Every point on the line represents a solution of the equation x + 2y = 7, and every solution of the equation corresponds to a point on the line.

Slide 3.2 - 5

The line gives a “picture” of all the solutions of the equation x + 2y = 7. Only a portion of the line is shown, but it extends indefinitely in both directions, suggested by the arrowheads.

The line is called the graph of the equation, and the process of plotting the ordered pairs and drawing the line through the corresponding points is called graphing.


Graph linear equations by plotting ordered pairs. (cont’d)

In summary, the graph of any linear equation in two variables is a straight line.

Slide 3.2 - 6

Since two distinct points determine a line, we can graph a straight line by finding any two different points on the line. However, it is a good idea to plot a third point as a check.

Notice the word line appears in the name “lineear equation.”


EXAMPLE 1 Graphing a Linear Equation

5 2 10.x y Solution: 105 2 0y

2 1

2 2

0y

When graphing a linear equation, all three points should lie on the same straight line. If they don’t, double-check the ordered pairs you found.

Slide 3.2 - 7

Graph

5y

05 2 10x 5 1

5 5

0x

2x

5 04 2 1y 2020 2 10 20y

2y

2 0

2 2

1y

02,

24,

0, 5


EXAMPLE 2 Graphing a Linear Equation

22.

3y x

Solution: 02

23

y

2y

Slide 3.2 - 8

Graph2

23

2 20 x

22

3

3 3

2 2x

3x

22

32 24 x

2

2 3

32 x

3x

3,0

43,

0, 2


Objective 22

Find intercepts.

Slide 3.2 - 9


In the previous example, the graph intersects (crosses) the y-axis at (0,−2) and the x-axis at (3,0). For this reason (0,−2) is called the y-intercept and (3,0) is called the x-intercept of the graph.

Find intercepts.

To find the x-intercept, let y = 0 and solve for x. Then (x,0) is the x-intercept.

The intercepts are particularly useful for graphing linear equations. The intercepts are found by replacing, in turn, each variable with 0 in the equation and solving for the value of the other variable.

Slide 3.2 - 10

To find the y-intercept, let x = 0 and solve for y. Then (0, y) is the y-intercept.


EXAMPLE 3

Solution:

Finding Intercepts

5 2 100x

Slide 3.2 - 11

Find the intercepts for 5x + 2y = 10. Then draw the graph.

5 0

5 5

1x

2x

5 0 2 10y 2 0

2 2

1y

5y

2,0 0,5y-intercept:x-intercept:

When choosing x- or y-values to find ordered pairs to plot, be careful to choose so that the resulting points are not too close together. This may result in an inaccurate line.


Objective 33

Graph linear equations of the form Ax + By = 0.

Slide 3.2 - 12


If A and B are nonzero real numbers, the graph of a linear equation of the form

passes through the origin (0,0).

Graph linear equations of the form Ax + By = 0.

0Ax By

Slide 3.2 - 13

A second point for a linear equation that passes through the origin can be found as follows:

1. Find a multiple of the coefficients of x and y.

2. Substitute this multiple for x.

3. Solve for y.

4. Use these results as a second ordered pair.


EXAMPLE 4

Solution:

Graphing an Equation of the Form Ax + By = 0

4 6 2 0y

Slide 3.2 - 14

Graph 4x − 2 = 0.

2424 242 0y 2 2

2 2

4y

12y

4 022x 44 44 0x

4 4

4 4

x

1x

112


Objective 44

Graph linear equations of the form y = k or x = k.

Slide 3.2 - 15


Graphing linear equations of the form y = k or x = k.

Slide 3.2 - 16

The equation y = −4 is the linear equation in which the coefficient of x is 0. Also, x = 3 is a linear equation in which the coefficient of y is 0. These equations lead to horizontal and vertical straight lines, respectively.

The graph of the linear equation y = k, where k is a real number, is a horizontal line with y-intercept (0, k) and no x-intercept.

The graph of the linear equation x = k, where k is a real number, is a vertical line with x-intercept (k ,0) and no y-intercept.

The equations of horizontal and vertical lines are often confused with each other. Remember that the graph of y = k is parallel to the x-axis and that of x = k is parallel to the y-axis.


EXAMPLE 5Graphing an Equation of the Form y = k

Solution:

The equation states that every value of y = −5.

Slide 3.2 - 17

Graph y = −5.


EXAMPLE 6

Solution:After 2 is added to each side the equation states thatevery value of x = 2.

Graphing an Equation of the Form x = k

Slide 3.2 - 18

Graph x − 2 = 0.


Objective 55

Use a linear equation to model data.

Slide 3.2 - 19


The different forms of linear equations from this section and the methods of graphing them are given in the following summary.

Slide 3.2 - 20


EXAMPLE 7

Solution:

a) about 525 billion dollars

Use a Linear Equation to Model Credit Card Debt

Slide 3.2 - 22

Use a) the graph and b) the equation to approximate credit card debt in 1997, where x = 2.

1997

38.7 452 0y

77.4 450y

527.4y

527.4 billion dollars.

b)

Chapter 3 Section 2 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.

Documents

Transcript of Chapter 3 Section 2 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.