���

Pre-Activity

PrePArAtion

Section 2.2 introduced validating ordered pair solutions to equations in two variables. This activity shows how to find ordered pair solutions to an equation in two variables by choosing one variable value and applying the general methodology for solving equations to find the value of the other variable. The following equations are a small sample of the widespread applications of finding ordered pair solutions to equations in two variables.

Food

You have $20 to buy the chips and salsa for a large party. If chips cost $3 and a jar of salsa costs $2, how many jars of salsa can you buy if you buy 4 bags of chips?

$3C + $2S = $20$3(4) + $2S = $20

Spor

ts In football, field goals score 3 points each and touchdowns score 7 points (with the extra point). If a team scores 2 field goals, how many touchdowns does it score if the total score is 48 points?

3F + 7T = 483(2) + 7T = 48

Fina

nce Money is divided between two accounts. One is low risk and

returns 6% interest. The other is higher risk and returns 11%. If you put $4000 in the low risk account, how much should you put in the high risk account to earn a total of $900 in interest?

0.11x + 0.06y = $9000.11x + 0.06(4000) = $900

Scie

nce

Converting between Fahrenheit and Celsius is common in Chemistry and Physics. The conversion formula is:

C F= -95

32( )

Convert 98.6 °F to °C:

C = -95

98 6 32( . )

• Find ordered pair solutions

• Find the x- and y-intercept of a line from its equation

• Find the slope of a line from its equation

• Calculate the slope of an equation, given any two ordered pairs

Ordered Pairs, Intercepts, and Slopes

Section 5.2

new terms to LeArn

slope

slope-intercept form

standard form

x-intercept

y-intercept

Previously used

coefficient

equation

integer

LeArning objectives

terminoLogy

linear equation

ordered pair solution

real numbers

��� Chapter � — Graph�ng

buiLding mAthemAticAL LAnguAge

Linear Equations

Equations in two variables are classified by the exponent (power) of the variables. First degree equations, that is, equations whose variables are raised to the first power, are linear equations. That means that when these equations are graphed in the Rectangular Coordinate System, their graphs make a straight line.

Linear equations fit a pattern that looks like Ax + By = C, where x and y are the variables and A, B, and C are real number constants with A and B not both equal to zero. Ax + By = C is called the standard form equation of a line.

For example, the equation 2x + 3y = 10 is a linear equation with infinitely many solutions; (2, 2), (5, 0), and (–1, 4) are just a few of them. You can see from the graph of this line (the figure at right) that solutions to a linear equation are points on the graph of that equation.

x and y-Intercepts

There are two ordered pair solutions that convey extra information and that have special names, the x- and y-intercepts. The x-intercept is the point at which the graph of the line crosses the x-axis. As you can see in the figure at right, that point is (5, 0). It is important to note that the y-coordinate of this point is zero. Similarly, the y-intercept is the point at which the graph of the line crosses the y-axis. The x-coordinate at this point is zero. Though we have not plotted this point on the graph of the line, you can see that it does exist at the point where the line crosses the y-axis.

The x and y-intercepts allow us to find two ordered pair solutions with minimal effort.

The point at which the line crosses the x-axis is the x-intercept at (x, 0). The point at which the line crosses the y-axis is the y-intercept at (0, y).

y = mx + b: The Slope-Intercept Form of the Equation of a Line

There is a special form of the equation for a line that allows us to read information directly from it. The slope-intercept form for the equation of a line looks like y = mx + b. From the slope-intercept form of an equation, we can determine two important pieces of information by inspection—just by reading the equation.

The first is m, the coefficient of x. That value has special meaning for the graph of the line. It is the slope of the line, or how steep or inclined the line is. The value b in the slope-intercept form indicates the second piece of information, the y-intercept, which is the point (0, b).

y

x(5, 0)

(-1, 4)(2, 2)

y-intercept

x-intercept

y = m x + by-interceptslope

���Sect�on �.� — Ordered Pa�rs, Intercepts, and Slopes

y

x(-1, 0)

(0, 2)

Linear Equation: –2x + y = 2 or y = 2x + 2x-intercept: (–1, 0) y-intercept: (0, 2) slope: 2

y

x(0, 1)

(1, 0)

Linear Equation: x + y = 1 or y = –x + 1x-intercept: (1, 0) y-intercept: (0, 1) slope: –1

Note that when the slope is a positive value, the line extends up, left to right. When the slope is a negative value, the slope extends down, left to right.

Calculating the Slope of a Line

It is possible to calculate the slope of a line, if you know two points on that line (i.e., two solutions to the equation of the line). It is not difficult, but does require that you understand what the slope of a line actually means. Think of a flight of stairs—each step consists of a riser (the vertical part) and the tread (the horizontal part). The flight of stairs consists of “rises” and “runs.” The slope of the stairs is a ratio of the rise or vertical change to the run or horizontal change.

You have already learned to calculate the length of both vertical and horizontal line segments; the slope is simply the vertical length divided by the horizontal length.

Let’s look at the slope of a line on the coordinate plane. (The equation for this line is: y = 2x – 5.)

In the figure at right, we can see that the rise is the vertical line segment that extends from –1 to 3 on the y-axis. We can calculate the length of this line as |y2 – y1| or |3 – (–1)| = 4.

The run is measured on the horizontal line segment that extends from 2 to 4 on the x-axis. We can calculate the length of this line as |x2 – x2| or |4 – 2| = 2.

The ratio of the rise to the run is

( )( )y yx x

2 1

2 1

--

, in this case 42

or 2. This is the slope of the line.

The Slope Formula allows us to perform these calculations in one neat step.

RUNhorizontal

(x-direction)

RISEvertical

(y-direction)

y

x

(2, -1)

(4, 3)(2, 3)

Slope Formula

Given the coordinates of any two points (x1, y1) and (x2, y2), the slope of their line is: m y y

x x=

--

( )( )

2 1

2 1

��0 Chapter � — Graph�ng

techniQues

2 Find the x- and y-Intercepts for a Linear Equation

1. To find an x-intercept, let y = 0 and solve for x.

2. To find a y-intercept, let x = 0 and solve for y.

3. Validate by substitution.

► Example 2: Find the x and y-intercepts for the equation 2x + 3y = 10

x-intercept: let y = 0 and solve for x:

2 +3(0)=10

2 =10

2 10=

2 2=5

x

x

x

x The x -intercept is (5, 0).

Validate: 2(5) + 3(0) ?= 10

10 = 10

y-intercept: let x = 0 and solve for y:

2(0)+3 =10

3 =10

3 10=

3 310

= 3

y

y

y

y10

The y -intercept is (0, ).3

Validate:

2(0) + 3(10

3)

?= 10

10 = 10

1 Find Ordered Pair Solutions to a Linear Equation

1. Choose any value for x (or y).

2. Solve for y (or x).

3. Validate by substituting the ordered pair back into the equation.

► Example 1: Find an ordered pair solution for the equation 2x + 3y = 10

Choose a value for x: Let x = –4

2 +3 =10

2(–4)+3 =10

–8+3 =10

+8–8+3 =10+8

3 =18

3 18=

3 3=6

x y

y

y

y

y

y

y

An ordered pair solution is: (–4, 6)

Validate: 2(–4) + 3(6) ?= 10

–8 + 18 ?= 10

10 = 10

or y: Let y = –2

2 +3 =10

2 +3(–2)=10

2 –6=10

2 –6+6=10+6

2 =16

2 16=

2 2=8

x y

x

x

x

x

x

xAn ordered pair solution is: (8, –2)

Validate: 2(8) + 3(–2) ?= 10

16 – 6 ?= 10

10 = 10

���Sect�on �.� — Ordered Pa�rs, Intercepts, and Slopes

3 Find the Slope of a Linear Equation

1. Find two ordered pair solutions to the equation (use Technique 1).

2. Substitute the ordered pair values into the Slope Formula to determine the slope.

3. Validate by finding a third ordered pair solution to the equation and using this point (together with one of the previous two ordered pair solutions) to recalculate the slope.

► Example 3: Find the slope for the equation 2x + 3y = 10.

Use the values previously calculated from Example 1:

(–4, 6) and (8, –2)

2 1

2 1

( – )m=

( – )

(–2–6)m=

(8–(–4))

(–8)m=

(12)

y y

x x

2=–

3slope

Validate: Choose a value for x: Let x = 2

An ordered pair solution is: (2, 2).

2 +3 =10

2(2)+3 =10

4+3 =10

4–4+3 =10–4

3 =6

=2

x y

y

y

y

y

y

Let (2, 2) be P1 and use (–4, 6) as P2. Substitute the values into the Slope Formula:

2 1

2 1

( – )m=

( – )

(6–2)m=

(–4–2)

(4)m=

(–6)

2= –

3slope

y y

x x

We can rearrange the original equation (2x + 3y = 10) into slope-intercept form and visually check the slope and y-intercept:

2 3 102 2 3 10 2

3 2 1033

23

103

23

103

x yx x y x

y xy x

y x

+ =

- + + = -

= - +

=-

+

= - +

y = m x + b

2 10=– +

3 3y x

��� Chapter � — Graph�ng

modeLs

Model 1

Find two ordered pair solutions for the equation x – 2y = 8:

Pick a value for x Pick a value for y

Let x = 4 Let y = 1

Solve for y Solve for x

x yy

yyy

y

- =

- =

- - = -

- =

--

=-

= -

2 84 2 84 4 2 8 4

2 422

42

2Ordered pair: (4, –2)

x yxxxx

- =

- =

- =

- + = +

=

2 82 1 82 82 2 8 210

( )

Ordered pair: (10, 1)

Validate Validate

x y- =

- - =

=

2 84 2 2 88 8

( )

?x y- =

- =

=

2 810 2 1 88 8

( )

?

Model 2

Find the x and y-intercepts for the equation x – 2y = 8:

x-intercept: let y = 0 and solve for x y-intercept: let x = 0 and solve for y

x yxxx

- =

- =

- =

=

2 82 0 80 88

( )

The x - intercept is (8, 0).

2 80 2 8

2 82 82 2

4

x yy

yy

y

− =− =

− =−

=− −= − The y - intercept is (0, – 4).

Validate Validate

?x y- =

- =

=

2 88 2 0 8

8 8( ) ( )

x y- =

- - =

=

2 80 2 4 8

8 8( ) ( )

?

���Sect�on �.� — Ordered Pa�rs, Intercepts, and Slopes

Model 3

Calculate the slope for the following equations:

Equation Find two ordered pair solutions Calculate the slope Validate

2x – y = 2 Let x = 1 2 1 22 2

00

( )- =

- =

- =

=

yy

yy

Ordered pair 1 (1, 0)

Let y = 2 2 2 22 4

2

xx

x

- =

=

=Ordered pair 2 (2, 2)

m y yx x

m

=--

=--

= =

( )( )( )( )

2 1

2 1

2 02 1

slope 21

2

Let x = 4 2 4 28 2

2 86

6

( )- =

- =

- = -

- = -

=

yy

yy

yOrdered pair (4, 6)

m y yx x

m

=--

=--

= =

( )( )( )( )

2 1

2 1

6 24 2

2slope 42

y = 3x + 2 Let x = 3 y xyy

= +

= +

= + =

3 23 3 29 2 11( )

Ordered pair 1 (3, 11)

Let y = 4 4 3 24 3 22 33 2

23

= +

= +

=

=

=

xxx

x

x

Ordered pair 2 2 ,43

2 1

2 1

( )( )(4 11)

2 33( 7)2 93 3

( 7)737 31 77 33 77 31 7

y ymx x

m

m

m

m

m

−=

−−

= −

−=

−

−=

− − − = − − = − −

= =21slope 37

Let x = 1 yyy

= +

= +

=

3 1 23 25

( )

Ordered pair (1, 5)

m y yx x

m

=--

=--

=--

= =

( )( )( )( )

( )( )

2 1

2 1

5 111 3

62

62

3slope

Though the mathematics for calculating the slope in the example above may seem overwhelming, remember that 1) it is only mathematics and 2) you are free to choose the value you want to use for x as you solve for y. There is something of an art to choosing values that keep fractions to a minimum; you will become increasingly skilled at this as you continue to work through this chapter.

��� Chapter � — Graph�ng

Model 4

What is the equation of a line with a slope of –2 and a y-intercept of (0, 4)? Use the slope-intercept form for your answer.

m = –2, b = 4, slope-intercept equation form: y = mx + b

Replace m and b with the given values: y = (–2)x + (4)

Slope-intercept form: y = –2x + 4

Model 5

Using the graph of the equation 2x + 2y = 6 shown at right, do the following:

a. Plot and label the ordered pair solutions: (0, 3), (1, 2), and (2, 1).

b. Label the x and y-intercepts.

c. Calculate the slope of the line. Validate your work.

The y-intercept was given. Find the x-intercept. Let y = 0 and solve for x:

2 2 0 62 6

62

3

xx

x

+ =

=

= =

( )

The x-intercept is (3, 0)

Use (0, 3) and (1, 2) to calculate the slope:

2 1

2 1

( )( )(2 3)(1 0)

1 1

y ymx x

m

−=

−−

=−

−= =slope –1

Validate:

Use (2, 1) and (0, 3) to recalculate the slope: 2 1

2 1

( )( )(3 1)(0 2)

2 2

y ymx x

m

−=

−−

=−

= =−

slope –1

y

x

y

x(3, 0)

x-intercept(2, 1)

(1, 2)

(0, 3)y-intercept

���Sect�on �.� — Ordered Pa�rs, Intercepts, and Slopes

Addressing common errors

Issue Incorrect Process Resolution Correct

Process Validation

Reversing coordinate values when using the Slope Formula

Given the points (1, 3) and (2, 7),

calculate the slope of their line.

m =--

= - = -

( )( )7 31 2

4slope 41

While either point can be P1, it is critical that you preserve that order when substituting values into the Slope Formula. If you find yourself making this error, consider labeling the points P1 and P2 to help you keep them straight.

P1 = (1, 3)P2 = (2, 7)

m y yx x

m

=--

=--

= =

( )( )( )( )

2 1

2 1

7 32 1

slope 441

Switch the points and recalculate:P1 = (2, 7), P2 = (1, 3)

m y yx x

m

=--

=--

=--

=

( )( )( )( )

2 1

2 1

3 71 2

4slope 41

Reporting a y-intercept as only a y-value rather than an ordered pair

Find the y-intercept:x + 4y = 12

Let x = 00 + 4y = 12

y = 3

The y-intercept is 3.

Remember that the intercepts are points, and therefore consist of both an x- and y-value, even when that value is zero.

Find the y-intercept:x + 4y = 12

Let x = 00 + 4y = 12

y = 3

The y-intercept is (0, 3).

You must substitute both the x- and y-values for any point, including intercepts, when validating your work.

(0) + 4(3) ?= 1212 = 12

Confusing the x- and y-coordinates in an ordered pair

If the x-coordinate is 3, what is the ordered pair solution for the equation x – y = 5?

x – 3 = 5x = 8(8, 3)

Solutions to equations are all about order—always associate x with the first coordinate value: (x, y).

If the x-coordinate is 3, the ordered pair will be (3, y). We must determine the value of y.x y

yy

yy

y

- =

- =

- - = -

- =

- - = -

= -

53 53 3 5 3

21 1 2

2( )( ) ( )

The ordered pair solution is (3, –2).

Substitute the ordered pair values into the original equation:

x – y = 5

(3) – (–2) ?= 5

3 + 2 ?= 55 = 5

��� Chapter � — Graph�ng

PrePArAtion inventory

Before proceeding, you should know each of the following:

What a linear equation is

How to find solutions to a linear equation

How to find the x- and y-intercepts of a linear equation

How to use the Slope Formula to calculate the slope of a line

How to rearrange a linear equation into the y = mx + b (slope-intercept) form

How to insert the slope and y-intercept into the y = mx + b (slope-intercept) form to create an equation for a line

Issue Incorrect Process Resolution Correct

Process Validation

Forgetting which intercept value is zero

Find the x- and y- intercepts of the

equation: 3x – y = 6

3(0) – y = 6y = – 63x – 0 = 63x = 6 x = 2

Answer:

The intercepts are (0, – 6) and (0, 2).

Label your work and record the ordered pairs as you work through a problem so that you do not forget which intercept you are finding.

Always validate ordered pair solutions.

Find the x- and y- intercepts of the

equation: 3x – y = 6

To find the y-intercept, let x = 0 and solve for y:

3(0) – y = 6y = – 6

The y-intercept is (0, – 6).

To find the x-intercept, let y = 0 and solve for x:

3x – 0 = 63x = 6 x = 2

The x-intercept is (2, 0).

For the y-intercept:3x – y = 6

3(0) – (–6) ?= 66 = 6

For the x-intercept:3x – y = 6

3(2) – 0 ?= 66 = 6

���

Activity

Section 5.2

PerformAnce criteriA

• Given an equation in two variables, be able to generate ordered pair solutions.– correct ordered pair form– correct x-coordinate– correct y-coordinate – correct validation

• Given the equation of a line, find the x- and y-intercepts.– correct ordered pair form– correct validation

• Given an equation in two variables, calculate the slope of the line representing that equation through use of the Slope Formula.– correct generation of ordered pair solutions– correct use of the Slope Formula– correct validation

• Given ordered pair solutions to a linear equation, calculate the slope of the line using the Slope Formula.– correct identification and ordering of points– correct calculations– correct validation

Ordered Pairs, Intercepts, and Slopes

criticAL thinking Questions

1. How do you know which values to pick when trying to find an ordered pair solution to an equation in two variables?

2. How do you validate an ordered pair solution?

3. What is the y-coordinate of the ordered pair solution to any linear equation at the x-intercept?

��� Chapter � — Graph�ng

4. What does it mean to “solve for y in terms of x?”

5. How can you determine the y-intercept of a line without rearranging its equation into y = mx + b (slope-intercept) form?

6. How might the graph of a line with a slope of 5 differ from the graph of a line with a slope of –5?

7. How might the graph of a line with a slope of 10 differ from the graph of a line with a slope of 110

?

tiPs for success

Document ordered pairs as you work through the problems: • Make sure ordered pairs are in the correct order (x-coordinate, y-coordinate)• Make sure that the zero coordinates of the intercepts are correctly noted.

���Sect�on �.� — Ordered Pa�rs, Intercepts, and Slopes

demonstrAte your understAnding

1. Find three ordered pair solutions for each of the following equations.

Problem Worked Solutions Validation (for each point)

a) 4x + 5y = 20

b) 3x – y = 6

�00 Chapter � — Graph�ng

2. Find the x- and y-intercepts for the following equations

Equation x-intercept y-intercept Validate

a) 4x – 2y = 6

b) 3x – y = 6

c) 6x + 3y = 18

d) 4x – 5y = 10

e) x – 2y = 4

�0�Sect�on �.� — Ordered Pa�rs, Intercepts, and Slopes

3. Calculate the slope for the following equations:

Equation Find two ordered pair solutions Calculate the slope Validate

a) 4x – 3y = 5

b) 6x + 3y = 18

�0� Chapter � — Graph�ng

4. Fill in the table below.

Standard Form Slope-Intercept Form Slope y-intercept

a) 23

(0, – 2)

b) – 1 (0, 3)

c) y = – 2x + 3

d) x – 2y = – 4

�0�Sect�on �.� — Ordered Pa�rs, Intercepts, and Slopes

5. Using the graph of the equation –x + 2y = 3 shown below, do the following:

a. Plot and label the ordered pair solutions: (–3, 0), (–1, 1) and (3, 3).b. Label the x and y-intercepts.c. Calculate the slope of the line. Validate your work.d. Complete the data fields at the bottom of the page.

y

x

Calculate the slope Validate

x-intercept: ____________ y-intercept: ____________ slope = ____________

�0� Chapter � — Graph�ng

In the second column, identify the error(s) in the worked solution or validate its answer. If the worked solution is incorrect, solve the problem correctly in the third column and validate your answer.

Worked Solution Identify Errors or Validate Correct Process Validation

1) If the y-coordinate is 15, what is the ordered pair solution for the equation 2x + 3y = 90?

2 15 3 9030 3 903 60

20

( )+ =

+ =

=

=

yy

yy

(15, 20)

2) Find the x-intercept:4x + y = –1

Let y = 0

4 0 114

x

x

+ = -

= -

3) Given the points (–6, 3) and (1, –1), calculate the slope

of their line.

m =- -- -

=

3 11 6

47

( )( )

slope

4) Find the y-intercept of the following equation:

x + y = 1

x yxx

+ =

+ =

=

10 11

The y-intercept is (1, 0).

identify And correct the errors