1. Interpret graphs.2. Write a solution as an ordered

pair.3. Decide whether a given ordered

pair is a solution of a given equation.

4. Complete ordered pairs for a given equation.

5. Complete a table of values.6. Plot ordered pairs.

Interpret Graphs.

A linear equation in two variables is an equation that can be written in the form

where A, B, and C are real numbers and A and B are not both 0.

Some examples of linear equations in two variables in this form, called standard form, are

and

Other linear equations in two variables, such as

and are not written in standard form, but could be. We discuss the forms of linear equations in more detail in Section 10.4.

,Ax B Cy

3 ,4 9x y ,0x y 8.2x y

4 5y x 3 ,7 2x y

Linear equations in two variables

10.1.2 Write a solution as an ordered pair.A solution of a linear equation in two variables requires two

numbers, one for each variable. For example, a true statement results when we replace x with 2 and y with 13 in the equation

since 4 5,y x 13 4 52 . The pair of numbers x = 2 and y = 13 gives a solution of the

equation The phrase “x = 2 and y = 13” is abbreviated 4 5.y x

2,13

x-value y-value

The x-value is always given first. A pair of numbers such as (2,13) is called an ordered pair, since the order in which the numbers are written is important.

The ordered pairs (2,13) and (13,2) are not the same. The second pair indicates that x = 13 and y = 2. For ordered pairs to be equal, their x-coordinates must be equal and their y-coordinates must be equal.

10.1.3 Decide whether a given ordered pair is a solution of a given equation.

We substitute the x- and y-values of an ordered pair into a linear equation in two variables to see whether the ordered pair is a solution. An ordered pair that is a solution of an equation is said to satisify the equation.

When listing ordered pairs, be sure to always list the x-value first.

Infinite numbers of ordered pairs can satisfy a linear equation in two variables.

Problem:

Decide (yes or no) whether each ordered pair is a solution of the equation

Solution:

4,20

5 2 20.x y

Deciding Whether Ordered Pairs Are Solutions of an Equation

2, 5 225 2 05

25 4 2 200

10 10 20 0 20

20 40 20 20 20

No

Yes

Complete each ordered pair for the equation

Solution:

2 9.y x

4 9y 22 9y

5y

8x

2, 2, 5

7 9 92 9x 16

2 2

2x

10.1.4 Complete ordered pairs for a given equation.

,7 8,7

Complete the table of values for the equation Then write the results as ordered

pairs. Solution:

2 3 12.x y

2 0 3 12y

10.1.5 Completing Tables of Value

3 12

3 3

y

4y

4

26

3

2

2 3 120x 2 2

2 2

1x

6x

2 3 3 12y

3

3 6

3

y

2y

63 12 66 y 132 3 2x

2 3

2 2

x

3

2x

99 12 92x

0,4

3, 2 32

,3

6,0

10.1.6 Plot ordered pairs.

Every linear in two variables equation has an infinite number of ordered pairs as solutions. Each choice of a number for one variable leads to a particular real number for the other variable.

To graph these solutions, represented as ordered pairs (x,y), we need two number lines, one for each variable. The two number lines are drawn as shown below. The horizontal number line is called the x-axis and the vertical line is called the y-axis. Together, these axes form a rectangular coordinate system, also called the Cartesian coordinate system.

Plot ordered pairs. (cont’d)The coordinate system is divided into four regions, called

quadrants. These quadrants are numbered counterclockwise, starting with the one in the top right quadrant.

Points on the axes themselves are not in any quadrant.

The point at which the x-axis and y-axis meet is called the origin, labeled 0 on the previous diagram. This is the point corresponding to (0, 0).

The x-axis and y-axis determine a plane— a flat surface illustrated by a sheet of paper. By referring to the two axes, we can associate every point in the plane with an ordered pair. The numbers in the ordered pair are called the coordinates of the point.

In a plane, both numbers in the ordered pair are needed to locate a point. The ordered pair is a name for the point.

Plot ordered pairs. (cont’d)

For example, locate the point associated with the ordered pair (2,3) by starting at the origin.

Since the x-coordinate is 2, go 2 units to the right along the x-axis.

Since the y-coordinate is 3, turn and go up 3 units on a line parallel to the y-axis.

The point (2,3) is plotted in the figure to the right. From now on the point with x-coordinate 2 and y-coordinate 3 will be referred to as point (2,3).

EXAMPLE:

Plot the given points in a coordinate system:

3,5 ,

Plotting Ordered Pairs

2,6 , 4,0 , 5, 2 , 5, 2 , 0, 6 .

EXAMPLE 7:

Complete the table of ordered pairs for the equation, , where x = year and y = number

of twin births in thousands. Round answers to the nearest whole number. Interpret the results for 2002.

3.563 7007.7.y x

Completing Ordered Pairs to Estimate the Number of Twin Births

20013.563 7007.7y 7.129.6 7007.7y 122y

19993.563 7007.7y 7122.4 7007.7y 115y

20023.563 7007.7y 7133.1 7007.7y 125y

115 122 125

Solution:

There were about 125 thousand twin births in the U.S. in 2002.

Plot ordered pairs. (cont’d)

The ordered pairs of twin births in the U.S. for 1998, 2000, and 2003 are graphed to the right. This graph of ordered pairs of data is called a scatter diagram. Notice how the how the axes are labeled: x represents the year, and y represents the number of twin births in thousands.

A scatter diagram enables us to tell whether two quantities are related to each other. These plotted points could be connected to form a straight line, so the variables x (years) and y (number of births have a linear relationship.

Plot ordered pairs. (cont’d)

Think of ordered pairs as representing an input value x and an output value y. If we input x into the equation, the output is y. We encounter many examples of this type of relationship every day.

The cost to fill a tank with gasoline depends on how many gallons are needed; the number of gallons is the input, and the cost is the output

The distance traveled depends on the traveling time; input a time and the output is a distance.

The growth of a plant depends on the amount of sun it gets; the input is the amount of sun, and the output is growth.

1. Graph linear equations by plotting ordered pairs.

2. Find intercepts.3. Graph linear equations of the form

Ax + By = 0.4. Graph linear equations of the form

y = k or x = k.5. Use a linear equation to model

data.

10.2 Graphing Linear Equations in Two Variables

10.2.1: 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.

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.

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.

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.”

Problem:

Graph

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.

5y

05 2 10x 5 1

5 5

0x

2x 02, 0, 5

Problem:

Graph

Graphing a Linear Equation

22.

3y x

Solution: 02

23

y

2y

22

32 20 x

22

3

3 3

2 2x

3x

22

32 24 x

2

2 3

32 x

3x

3,0

43,

0, 2

10.2.2: Find intercepts.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.

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.

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

Problem:

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

Finding Intercepts

5 2 100x 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.

10.2.3: Graph linear equations of the formAx + By = 0.

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

passes through the origin (0,0).

0Ax By

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.

Problem:

Graph 4x − 2y = 0.Solution:

Graphing an Equation of the Form Ax + By = 0

4 6 2 0y 2424 242 0y 2 2

2 2

4y

12y

4 022x 44 44 0x

4 4

4 4

x

1x

112

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

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.

Problem:

Graph y = −5.

Graphing an Equation of the Form y = k

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

Problem:

Graph x − 2 = 0.Solution:After 2 is added to each side the equation states thatevery value of x = 2.

Graphing an Equation of the Form x = k

10.2.5: Use a linear equation to model data.

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

Problem:

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

Solution:a) about 525 billion dollars

Use a Linear Equation to Model Credit Card Debt

1997

38.7 452 0y

77.4 450y

527.4y 527.4 billion dollars.

b)

1. Find the slope of a line given two points.

2. Find the slope from the equation of a line.

3. Use slopes to determine whether two lines are parallel, perpendicular, or neither.

10.3: The Slope of a Line

An important characteristic of the lines we graphed in Section 10.2 is their slant, or “steepness.”

One way to measure the steepness of a line is to compare the vertical change in the line with the horizontal change while moving along the line from one fixed point to another. This measure of steepness is called the slope of the line.

The Slope of a Line

To find the steepness, or slope, of the line in the figure below, begin at point Q and move to point P. The vertical change, or rise, is the change in the y-values, which is the difference 6 − 1 = 5 units. The horizontal change, or run, is the change in the x-values, which is the difference 5 − 2 = 3 units.

10.3.1: Find the slope of a line given two points.

vertical change in rise

horizontal change in run 3e

5slop

x

y

The slope is the ratio of the vertical change in y to the horizontal change in x.

Count squares on the grid to find the change. Upward and rightward movements are positive. Downward and leftward movements are negative.

Problem:Finding the Slope of a Line

Solution:

Find the slope of the line.

6

1m

6m

The slope of a line is the same for any two points on the line.The slope of a line can be found through two nonspecific

points. This notation is called subscript notation, read x1 as “x-sub-one” and x2 as “x-sub-two”.

Find the slope of a line given two points. (cont’d)

21 2

2 1

1

horizontal change in run

vertical change iif

n ris .

e

y yx x

x x x

ym

Traditionally, the letter m represents slope.The slope of a line through the points (x1, y1) and (x2, y2) is

Moving along the line from the point (x1, y1) to the point (x2, y2), we see that y changes by y2 − y1 units. This is the vertical change (rise). Similarly, x changes by x2 − x1 units, which is the horizontal change (run). The slope of the line is the ratio of y2 − y1 to x2 − x1.

Problem:

2 1

2 1x

y y

x

Solution:

Finding Slopes of Lines

4 8

2 6m

12

8

3

2

and yield the same slope. Make sure to start with the

x- and y-values of the same point and subtract the x- and y-values of the other point.

Find the slope of the line through (6, −8) and (−2,4).

1 2

1 2x

y y

x

Positive and Negative Slopes

A line with a positive slope rises (slants up) from left to right.

A line with a negative slope falls (slants down) from left to right.

Find the slope of a line given two points. (cont’d)

Slopes of Horizontal and Vertical LinesHorizontal lines, with equations of the form y = k, have

slope 0.

Vertical lines, with equations of the form x = k, have undefined slopes.

Problem:

Solution:

5 5

1 2m

0

3

0

Find the slope of the line through (2, 5) and (−1,5).

Finding the Slope of a Horizontal Line

Problem:

Solution:

Finding the Slope of a Vertical Line

3 3

4 1m

5

0

Find the slope of the line through (3, 1) and (3,−4).

undefined slope

Consider the equation y = −3x + 5.

The slope of the line can be found by choosing two different points for value x and then solving for the corresponding values of y. We choose x = −2 and x = 4.

3 5y x

7y

3 5y x

11y

10.3.2: Find the slope from the equation of a line.

3 52y 6 5y

3 4 5y 12 5y

The ordered pairs are (−2,11) and (4, −7). Now we use the slope formula.

11 7 18

63

2 4m

Find the slope from the equation of a line. (cont’d)

Step 1: Solve the equation for y.

Step 2: The slope is given by the coefficient of x.

The slope, −3 is found, which is the same number as the coefficient of x in the given equation y = −3x + 5. It can be shown that this always happens , as long as the equation is solved for y.

This fact is used to find the slope of a line from its equation, by:

Problem:

Solution:

Finding Slopes from Equations

33 32 9xyx x

Find the slope of the line 3x + 2y = 9.

3 9

2 2y x

3

2m

2 3

2 2

9y x

Two lines in a plane that never intersect are parallel. We use slopes to tell whether two lines are parallel. Nonvertical parallel lines always have equal slopes.

10.3.3: Use slopes to determine whether two lines are parallel, perpendicular, or neither.

1

a

Lines are perpendicular if they intersect at a 90° angle. The product of the slopes of two perpendicular lines, neither of which is vertical, is always −1. This means that the slopes of perpendicular lines are negative (or opposite) reciprocals—if one slope is the nonzero number a, the other is . The table to the right shows several examples.

Problem:Deciding whether Two Lines Are Parallel or Perpendicular

Solution:3 4x y 3 9x y

3 33 4x xyx 3 9xy xx 4 3

1 1

xy

4 3y x

3 9

3 3 3

xy

13

3y x

13 1

3

3m 1

3m

The product of their slopes is −1, so they are perpendicular

Determine whether the pair of lines is parallel, perpendicular, or neither.

1. Write an equation of a line by using its slope and y-intercept.

2. Graph a line by using its slope and a point on the line.

3. Write an equation of a line by using its slope and any point on the line.

4. Write an equation of a line by using two points on the line.

5. Find an equation of a line that fits a data set.

10.4: Equations of a Line

10.4.1: Write an equation of a line by using its slope and y-intercept.

Problem:

Find an equation of the line with slope −1 and y-intercept (0,8).

Finding an Equation of a Line

Solution: 8y x

10.4.2: Graph a line by using its slope and a point on the line.

Step 1: Write the equation in slope-intercept form, if necessary, by solving for y.

Step 2: Identify the y-intercept. Graph the point (0,b).

Step 3: Identify slope m of the line. Use the geometric interpretation of slope (“rise over run”) to find another point on the graph by counting from the y-intercept.

Step 4: Join the two points with a line to obtain the graph.

Problem:

Solution:

Graphing a Line by Using the Slope and y-intercept

33 34 8xyx x

Graph 3x – 4y = 8 by using the slope and y-intercept.

4 3

4 4 4

8xy

32

4y x

Slope intercept form

Problem:

Solution:

Graphing a Line by Using the Slope and a Point

Graph the line through (2,−3) with slope .1

3

Make sure when you begin counting for a second point you begin at the given point, not at the origin.

10.4.3: Write an equation of a line by using its slope and any point on the line.

There is another form that can be used to write the equation of a line. To develop this form, let m represent the slope of a line and let (x1,y1) represent a given point on the line. Let (x, y) represent any other point on the line.

The point-slope form of the equation of a line with slope m passing through point (x1,y1) is

This result is the point-slope form of the equation of a line.

11 1

1

m xy

xy

xx

xx

11 yx xm y

1 1my xy x

Slope

Given point

1

1

my y

x x

Problem:

Solution:

Using the Slope-Intercept Form to Write an Equation

y m bx

Write an equation, in slope-intercept form, of the line having slope −2 and passing through the point (−1,4).

4 2 1 b 4 22 2b

2b

2 2y x The slope-intercept form is

Problem:

Solution:

Using the Point-Slope Form to Write Equations

Find the equation of the line through (5,2), with the slope Give the final answer in slope-intercept form.

1.

3

1 1my xy x

12 5

3y x

1 52

3

6

32

3y x

1 11

3 3xy

Problem:

Solution:

10.4.4: Finding the Equation of a Line by Using Two Points

2 1

2 1x

ym

y

x

Find an equation of the line through the points (2,5) and (−1,6). Give the final answer in slope-intercept form.

The same result would also be found by substituting the slope and either given point in slope-intercept form and then solving for b.

6 5

1 2m

1

3m

1 1my xy x

1

36 1xy

1 17

3 3xy

1 16

3

18

36

3xy

Standard Form

Many of the linear equations in Section 10.1−10.3 were given in the form Ax + By = C, called standard form, which we define as

follows.

A linear equation is in standard form if it is written as where A, B, and C are integers, A > 0, and B ≠ 0.

,Ax B Cy

A summary of the forms of linear equations follow

Problem:

Solution:

10.4.5: Finding the Equation of a Line That Describes Data

Use the points (1, 3362) and (7, 5491) to find an equation in slope-intercept form that approximates the data of the table. (Round the slope to the nearest tenth.) How well does this equation approximate the cost in year 5?

2 1

2 1x

ym

y

x

5491 3362

7 1m

9

6

212m

1 1my xy x 35 .362 183 4y x

354.8 3007.2xy 3362 354.83362 33.8 62354xy

354.8m The equation gives y ≈ 4781 when x = 5, which is a

pretty good approximation.

Competition Problems

Find the value of y if the line that contains (3, 4) and (4, 2) is perpendicular to the line that contains (6,y) and (22,3).

Answer:

-5

Write an equation in standard form of the line that is perpendicular to the line x + 2y = 10 and

has the same y intercept as the line

y – 1/2x = 3

Answer:

2x – y = -3

Write the equation of the line that passes through the point (1, 2) which has the same slope as the line through (4, -1) and (3, 6).

Answer:

y = -7x + 9

Find the equation of the line that passes through the points

(3,7) and (10,21).

Answer:

y = 2x + 1

Choose the equation of a line perpendicular to the graph -9x + 3y -12 = 0.

(a) y = -⅓x + 4

(b) y = -3x + 4

(c) y = -x + 1

(d)y = 3x + 5

Answer:

(a) y = -⅓x + 4

The graph of the equation

2x + 6y = 1

does not cross which quadrant?

Answer:

III

Find the equation in standard form of the line having slope of -1/4

and y-intercept of 10.

Answer:

A line passes through the points

(3, –4) and (–2, 5).

What is the sum of its

slope and y-intercept?

Answer:

-2/5

Find the slope of the line passing through (-5, 8) and (3, -2).

Answer:

-5/4

Write an equation of the line containing the given point and

perpendicular to the given line.

(-6,2) & 3x -9y = 2

Answer:

y = -3x - 16

Give the equation of a line, in standard form, that is

perpendicular to the graph of ¾x - ¼y = -½

and passes through the point

(2, -3).

Answer:

7x + 21y = -9