what is the answer to lesson 3.2 worksheet slope by equation

Learning Objectives

Past the terminate of this department, you lot will be able to:

• Find the slope of a line
• Graph a line given a point and the gradient
• Graph a line using its slope and intercept
• Cull the most convenient method to graph a line
• Graph and interpret applications of slope–intercept
• Use slopes to identify parallel and perpendicular lines

Be Prepared three.4

Before yous get started, take this readiness quiz.

Simplify: $\frac{\left(\mathrm{ane}–4\right)}{\left(8-2\right)}.$
If you missed this problem, review Instance 1.30.

Be Prepared 3.v

Dissever: $\frac{0}{4},\frac{\mathrm{iv}}{0}.$
If y'all missed this problem, review Example 1.49.

Be Prepared iii.6

Simplify: $\frac{15}{\mathrm{-3}},\frac{\mathrm{-15}}{\mathrm{iii}},\frac{\mathrm{-xv}}{\mathrm{-three}}.$
If you missed this problem, review Instance 1.30.

Observe the Gradient of a Line

When y'all graph linear equations, yous may notice that some lines tilt up every bit they go from left to right and some lines tilt downwardly. Some lines are very steep and some lines are flatter.

In mathematics, the measure out of the steepness of a line is called the gradient of the line.

The concept of slope has many applications in the real world. In construction the pitch of a roof, the slant of the plumbing pipes, and the steepness of the stairs are all applications of slope. and as you ski or jog downwards a hill, you lot definitely experience slope.

We can assign a numerical value to the slope of a line by finding the ratio of the rising and run. The ascension is the amount the vertical distance changes while the run measures the horizontal modify, as shown in this illustration. Slope is a charge per unit of change. See Figure 3.5.

Effigy three.v

Slope of a Line

The slope of a line is $\mathrm{thousand}=\frac{\text{rise}}{\text{run}}.$

The rise measures the vertical change and the run measures the horizontal change.

To find the slope of a line, we locate ii points on the line whose coordinates are integers. Then we sketch a right triangle where the two points are vertices and one side is horizontal and i side is vertical.

To find the slope of the line, we measure the distance along the vertical and horizontal sides of the triangle. The vertical distance is called the ascent and the horizontal altitude is chosen the run,

How To

Find the slope of a line from its graph using $m=\frac{\text{rise}}{\text{run}}.$

1. Pace 1. Locate two points on the line whose coordinates are integers.
2. Pace 2. Starting with one betoken, sketch a right triangle, going from the first point to the 2d point.
3. Footstep three. Count the rise and the run on the legs of the triangle.
4. Step 4. Take the ratio of rise to run to observe the gradient: $\mathrm{thousand}=\frac{\text{rise}}{\text{run}}.$

Example three.12

Find the slope of the line shown.

Try It 3.23

Find the slope of the line shown.

Effort It iii.24

Observe the gradient of the line shown.

How practise nosotros detect the gradient of horizontal and vertical lines? To find the slope of the horizontal line, $y=4,$ nosotros could graph the line, notice two points on it, and count the rise and the run. Let'south see what happens when we do this, as shown in the graph below.

What is the rise?	The rise is 0.
What is the run?	The run is 3.
What is the slope?	$\mathrm{1000}=\frac{\text{ascent}}{\text{run}}$
	$m=\frac{0}{3}$
	$\mathrm{one\; thousand}=0$
	The slope of the horizontal line $y=\mathrm{four}\text{is 0.}$

Let's also consider a vertical line, the line $\mathrm{10}=3,$ as shown in the graph.

What is the rise?	The rise is 2.
What is the run?	The run is 0.
What is the slope?	$\mathrm{thou}=\frac{\text{rise}}{\text{run}}$
	$m=\frac{2}{0}$

The gradient is undefined since sectionalisation by zero is undefined. And so we say that the slope of the vertical line $x=3$ is undefined.

All horizontal lines have slope 0. When the y-coordinates are the same, the rise is 0.

The slope of whatsoever vertical line is undefined. When the x-coordinates of a line are all the aforementioned, the run is 0.

Slope of a Horizontal and Vertical Line

The slope of a horizontal line, $y=b,$ is 0.

The gradient of a vertical line, $\mathrm{10}=a,$ is undefined.

Instance iii.13

Find the slope of each line: ⓐ $x=8$ ⓑ $y=\mathrm{-5}.$

Try It 3.25

Observe the gradient of the line: $x=\mathrm{-iv}.$

Attempt It three.26

Find the slope of the line: $y=7.$

Quick Guide to the Slopes of Lines

Sometimes we'll need to detect the gradient of a line betwixt 2 points when nosotros don't have a graph to count out the rising and the run. We could plot the points on grid paper, then count out the ascent and the run, simply as we'll see, there is a way to find the slope without graphing. Before we go to it, we need to innovate some algebraic notation.

We accept seen that an ordered pair $\left(x,y\right)$ gives the coordinates of a indicate. Simply when nosotros work with slopes, we utilise two points. How can the aforementioned symbol $\left(x,y\right)$ be used to stand for two dissimilar points? Mathematicians employ subscripts to distinguish the points.

$$\begin{array}{cc}\left(x_1,y_1\right)\hfill & \text{read "}x\text{sub one,}y\text{sub i"}\hfill \\ \left(x_2,y_{\mathrm{two}}\right)\hfill & \text{read "}x\text{sub 2,}y\text{sub 2"}\hfill \end{array}$$

We will employ $\left(x_1,y_1\right)$ to place the first point and $\left(x_{\mathrm{ii}},y_{\mathrm{ii}}\right)$ to identify the 2d point.

If nosotros had more than two points, we could use $\left(x_3,y_{\mathrm{iii}}\right),$ $\left(x_4,y_{\mathrm{iv}}\right),$ and then on.

Let'south meet how the rise and run relate to the coordinates of the two points by taking another await at the gradient of the line between the points $\left(\mathrm{ii},3\right)$ and $\left(7,\mathrm{vi}\right),$ every bit shown in this graph.

Since nosotros have two points, we will utilise subscript notation.	$\left(\stackrel{{\mathrm{10}}_{\mathrm{one}},}{2,}\stackrel{y_1}{\mathrm{three}}\right)\left(\stackrel{x_{\mathrm{two}},y_2}{7,6}\right)$
	$m=\frac{\text{rise}}{\text{run}}$
On the graph, nosotros counted the rise of 3 and the run of 5.	$\mathrm{one\; thousand}=\frac{3}{\mathrm{v}}$
Notice that the ascent of 3 can be found by subtracting the y-coordinates, 6 and 3, and the run of 5 tin can be found by subtracting the x-coordinates 7 and ii.
We rewrite the ascent and run by putting in the coordinates.	$m=\frac{6-3}{\mathrm{vii}-2}$
But 6 is $y_2,$ the y-coordinate of the second point and 3 is $y_{\mathrm{ane}},$ the y-coordinate of the first point. So we can rewrite the slope using subscript notation.	$m=\frac{y_2-y_1}{7-2}$
Also 7 is the x-coordinate of the second point and two is the x-coordinate of the start point. Then once again we rewrite the slope using subscript annotation.	$\mathrm{1000}=\frac{y_{\mathrm{ii}}-y_{\mathrm{ane}}}{x_2-x_1}$

We've shown that $\mathrm{1000}=\frac{y_{\mathrm{ii}}-y_{\mathrm{ane}}}{{\mathrm{ten}}_2-x_1}$ is actually another version of $m=\frac{\text{ascension}}{\text{run}}.$ We tin can use this formula to notice the slope of a line when we accept two points on the line.

Slope of a line between two points

The slope of the line between two points $(x_{\mathrm{i}},y_{\mathrm{one}})$ and $({\mathrm{10}}_{\mathrm{ii}},y_2)$ is:

$$m=\frac{y_2-y_1}{x_2-{\mathrm{ten}}_1}.$$

The slope is:

$$\begin{array}{c}y\text{of the 2d indicate minus}y\text{of the offset point}\hfill \\ \hfill \text{over}\hfill \\ x\text{of the 2nd indicate minus}x\text{of the outset point.}\hfill \end{array}$$

Example 3.14

Use the slope formula to discover the gradient of the line through the points $\left(\mathrm{-2},\mathrm{-3}\right)$ and $\left(\mathrm{-7},4\right).$

Try It 3.27

Utilise the slope formula to find the slope of the line through the pair of points: $\left(\mathrm{-iii},\mathrm{iv}\right)$ and $\left(2,\mathrm{-ane}\right).$

Endeavor It 3.28

Utilise the slope formula to detect the slope of the line through the pair of points: $\left(\mathrm{-two},6\right)$ and $\left(\mathrm{-iii},\mathrm{-4}\right).$

Graph a Line Given a Point and the Slope

Up to now, in this chapter, we have graphed lines by plotting points, by using intercepts, and past recognizing horizontal and vertical lines.

We can also graph a line when nosotros know one indicate and the slope of the line. We will first by plotting the point and and so utilise the definition of slope to draw the graph of the line.

Case 3.15

How to graph a Line Given a Point and the Slope

Graph the line passing through the point $\left(1,\mathrm{-ane}\right)$ whose slope is $\mathrm{grand}=\frac{\mathrm{three}}{4}.$

Try Information technology 3.29

Graph the line passing through the point $\left(2,\mathrm{-2}\right)$ with the slope $m=\frac{4}{3}.$

Try It 3.30

Graph the line passing through the point $\left(\mathrm{-2},3\right)$ with the slope $m=\frac{1}{\mathrm{four}}.$

How To

Graph a line given a signal and the slope.

1. Stride 1. Plot the given point.
2. Stride 2. Use the slope formula $\mathrm{1000}=\frac{\text{rise}}{\text{run}}$ to place the rising and the run.
3. Step 3. Starting at the given betoken, count out the rise and run to mark the second indicate.
4. Step 4. Connect the points with a line.

Graph a Line Using its Slope and Intercept

We have graphed linear equations by plotting points, using intercepts, recognizing horizontal and vertical lines, and using one point and the slope of the line. Once we see how an equation in slope–intercept grade and its graph are related, nosotros'll have one more than method we can use to graph lines.

See Figure 3.half dozen. Allow's await at the graph of the equation $y=\frac{1}{2}x+\mathrm{three}$ and observe its slope and y-intercept.

Figure 3.6

The carmine lines in the graph testify us the rise is 1 and the run is two. Substituting into the slope formula:

$$\begin{array}{}\\ \\ m=\frac{\text{rise}}{\text{run}}\hfill \\ m=\frac{\text{1}}{\text{ii}}\hfill \end{array}$$

The y-intercept is $\left(0,3\right).$

Look at the equation of this line.

Expect at the slope and y-intercept.

When a linear equation is solved for y, the coefficient of the x term is the slope and the constant term is the y-coordinate of the y-intercept. We say that the equation $y=\frac{\mathrm{i}}{2}x+3$ is in gradient–intercept form. Sometimes the slope–intercept class is called the "y-grade."

Slope Intercept Form of an Equation of a Line

The slope–intercept form of an equation of a line with slope m and y-intercept, $\left(0,b\right)$ is $y=mx+b.$

Let'southward practice finding the values of the slope and y-intercept from the equation of a line.

Example 3.16

Identify the gradient and y-intercept of the line from the equation:

ⓐ $y=-\frac{4}{\mathrm{seven}}\mathrm{ten}-2$ ⓑ $\mathrm{10}+3y=\mathrm{nine}$

Attempt It 3.31

Identify the slope and y-intercept from the equation of the line.

ⓐ $y=\frac{2}{\mathrm{five}}x-\mathrm{one}$ ⓑ $\mathrm{10}+4y=8$

Try Information technology 3.32

Identify the slope and y-intercept from the equation of the line.

ⓐ $y=-\frac{\mathrm{iv}}{\mathrm{three}}\mathrm{10}+\mathrm{one}$ ⓑ $3x+2y=12$

Nosotros accept graphed a line using the slope and a point. Now that we know how to find the slope and y-intercept of a line from its equation, we tin use the y-intercept every bit the point, and so count out the gradient from there.

Example 3.17

Graph the line of the equation $y=\text{−}x+4$ using its slope and y-intercept.

Attempt It 3.33

Graph the line of the equation $y=\text{−}x-3$ using its slope and y-intercept.

Attempt Information technology 3.34

Graph the line of the equation $y=\text{−}\mathrm{ten}-1$ using its slope and y-intercept.

Now that nosotros have graphed lines by using the slope and y-intercept, let's summarize all the methods nosotros have used to graph lines.

Cull the Most Convenient Method to Graph a Line

Now that we take seen several methods we can use to graph lines, how do we know which method to use for a given equation?

While we could plot points, use the slope–intercept course, or notice the intercepts for whatever equation, if nosotros recognize the near convenient way to graph a certain type of equation, our work will exist easier.

Generally, plotting points is non the near efficient way to graph a line. Let'due south look for some patterns to help determine the most user-friendly method to graph a line.

Here are 5 equations we graphed in this chapter, and the method we used to graph each of them.

$$\begin{array}{ccccc}& \mathbf{\text{Equation}}\hfill & & & \mathbf{\text{Method}}\hfill \\ \text{\#1}\hfill & x=2\hfill & & & \text{Vertical line}\hfill \\ \text{\#two}\hfill & y=\mathrm{-1}\hfill & & & \text{Horizontal line}\hfill \\ \text{\#3}\hfill & \text{−}x+2y=\mathrm{six}\hfill & & & \text{Intercepts}\hfill \\ \text{\#4}\hfill & 4\mathrm{10}-3y=12\hfill & & & \text{Intercepts}\hfill \\ \text{\#5}\hfill & y=\text{−}x+4\hfill & & & \text{Gradient–intercept}\hfill \end{array}$$

Equations #1 and #2 each accept just one variable. Remember, in equations of this form the value of that one variable is constant; it does not depend on the value of the other variable. Equations of this form accept graphs that are vertical or horizontal lines.

In equations #three and #iv, both x and y are on the same side of the equation. These two equations are of the class $A\mathrm{ten}+By=C.$ Nosotros substituted $y=0$ to observe the x- intercept and $x=0$ to find the y-intercept, and then found a third point by choosing some other value for ten or y.

Equation #5 is written in slope–intercept form. After identifying the gradient and y-intercept from the equation we used them to graph the line.

This leads to the following strategy.

Strategy for Choosing the Most Convenient Method to Graph a Line

Consider the form of the equation.

• If it only has 1 variable, it is a vertical or horizontal line.
  • $x=a$ is a vertical line passing through the ten-axis at a.
  • $y=b$ is a horizontal line passing through the y-axis at b.
• If y is isolated on one side of the equation, in the form $y=mx+b,$ graph by using the slope and y-intercept.
  • Identify the slope and y-intercept and and so graph.
• If the equation is of the course $A\mathrm{10}+By=C,$ discover the intercepts.
  • Find the x- and y-intercepts, a third bespeak, and then graph.

Example 3.18

Determine the most user-friendly method to graph each line:

ⓐ $y=5$ ⓑ $4x-5y=\mathrm{xx}$ ⓒ $x=\mathrm{-3}$ ⓓ $y=-\frac{\mathrm{five}}{\mathrm{ix}}\mathrm{10}+8$

Try It iii.35

Make up one's mind the nearly convenient method to graph each line:

ⓐ $3\mathrm{10}+\mathrm{ii}y=12$ ⓑ $y=4$ ⓒ $y=\frac{1}{5}\mathrm{10}-\mathrm{four}$ ⓓ $x=\mathrm{-vii}.$

Try It iii.36

Make up one's mind the most convenient method to graph each line:

ⓐ $\mathrm{10}=\mathrm{six}$ ⓑ $y=-\frac{3}{4}x+\mathrm{one}$ ⓒ $y=\mathrm{-viii}$ ⓓ $4x-3y=\mathrm{-1}.$

Graph and Translate Applications of Slope–Intercept

Many existent-world applications are modeled by linear equations. We volition accept a look at a few applications here and so you tin see how equations written in slope–intercept course relate to real globe situations.

Usually, when a linear equation models uses real-world data, different letters are used for the variables, instead of using but ten and y. The variable names remind us of what quantities are being measured.

Also, we often will demand to extend the axes in our rectangular coordinate organisation to bigger positive and negative numbers to accommodate the data in the awarding.

Example three.19

The equation $F=\frac{9}{5}C+32$ is used to convert temperatures, C, on the Celsius calibration to temperatures, F, on the Fahrenheit scale.

ⓐ Find the Fahrenheit temperature for a Celsius temperature of 0.

ⓑ Find the Fahrenheit temperature for a Celsius temperature of twenty.

ⓒ Interpret the slope and F-intercept of the equation.

ⓓ Graph the equation.

Try It iii.37

The equation $h=2\mathrm{south}+50$ is used to gauge a woman's height in inches, h, based on her shoe size, s.

ⓐ Estimate the tiptop of a child who wears women's shoe size 0.

ⓑ Gauge the top of a adult female with shoe size 8.

ⓒ Interpret the slope and h-intercept of the equation.

ⓓ Graph the equation.

Attempt Information technology iii.38

The equation $T=\frac{\mathrm{one}}{\mathrm{iv}}n+\mathrm{xl}$ is used to estimate the temperature in degrees Fahrenheit, T, based on the number of cricket chirps, n, in i infinitesimal.

ⓐ Estimate the temperature when there are no chirps.

ⓑ Estimate the temperature when the number of chirps in one minute is 100.

ⓒ Translate the slope and T-intercept of the equation.

ⓓ Graph the equation.

The cost of running some types of business has ii components—a stock-still toll and a variable cost. The fixed cost is always the aforementioned regardless of how many units are produced. This is the cost of rent, insurance, equipment, advertising, and other items that must be paid regularly. The variable cost depends on the number of units produced. It is for the material and labor needed to produce each item.

Example 3.20

Sam drives a delivery van. The equation $C=\mathrm{0.v}m+\mathrm{lx}$ models the relation between his weekly toll, C, in dollars and the number of miles, m, that he drives.

ⓐ Find Sam's cost for a week when he drives 0 miles.

ⓑ Find the price for a week when he drives 250 miles.

ⓒ Interpret the slope and C-intercept of the equation.

ⓓ Graph the equation.

Try Information technology 3.39

Stella has a abode business selling gourmet pizzas. The equation $C=4p+25$ models the relation between her weekly cost, C, in dollars and the number of pizzas, p, that she sells.

ⓐ Discover Stella's cost for a week when she sells no pizzas.

ⓑ Find the price for a week when she sells 15 pizzas.

ⓒ Interpret the slope and C-intercept of the equation.

ⓓ Graph the equation.

Try It 3.40

Loreen has a calligraphy concern. The equation $C=1.8n+35$ models the relation betwixt her weekly cost, C, in dollars and the number of nuptials invitations, n, that she writes.

ⓐ Find Loreen's cost for a week when she writes no invitations.

ⓑ Find the cost for a calendar week when she writes 75 invitations.

ⓒ Interpret the slope and C-intercept of the equation.

ⓓ Graph the equation.

Use Slopes to Identify Parallel and Perpendicular Lines

Two lines that have the same slope are chosen parallel lines. Parallel lines take the aforementioned steepness and never intersect.

We say this more formally in terms of the rectangular coordinate organisation. Ii lines that have the same gradient and different y-intercepts are called parallel lines. Run into Figure iii.7.

Figure 3.7

Verify that both lines accept the aforementioned gradient, $k=\frac{2}{\mathrm{v}},$ and different y-intercepts.

What about vertical lines? The slope of a vertical line is undefined, so vertical lines don't fit in the definition higher up. We say that vertical lines that have different ten-intercepts are parallel, like the lines shown in this graph.

Figure 3.8

Parallel Lines

Parallel lines are lines in the same aeroplane that exercise non intersect.

• Parallel lines have the aforementioned slope and dissimilar y-intercepts.
• If $m_1$ and $m_2$ are the slopes of 2 parallel lines then $m_{\mathrm{ane}}={\mathrm{thousand}}_2.$
• Parallel vertical lines have different x-intercepts.

Since parallel lines have the same gradient and different y-intercepts, we can at present just wait at the slope–intercept form of the equations of lines and decide if the lines are parallel.

Instance 3.21

Use slopes and y-intercepts to determine if the lines are parallel:

ⓐ $\mathrm{three}x-\mathrm{two}y=6$ and $y=\frac{\mathrm{three}}{2}x+\mathrm{i}$ ⓑ $y=2x-\mathrm{iii}$ and $\mathrm{-6}x+3y=\mathrm{-9}.$

Endeavour It three.41

Employ slopes and y-intercepts to determine if the lines are parallel:

ⓐ $2x+\mathrm{five}y=5$ and $y=-\frac{2}{\mathrm{five}}\mathrm{ten}-4$ ⓑ $y=-\frac{1}{2}x-\mathrm{one}$ and $\mathrm{10}+\mathrm{ii}y=\mathrm{-2}.$

Endeavor Information technology 3.42

Apply slopes and y-intercepts to decide if the lines are parallel:

ⓐ $4x-\mathrm{three}y=6$ and $y=\frac{4}{\mathrm{iii}}\mathrm{10}-1$ ⓑ $y=\frac{3}{4}\mathrm{ten}-3$ and $3\mathrm{10}-4y=12.$

Example 3.22

Apply slopes and y-intercepts to decide if the lines are parallel:

ⓐ $y=\mathrm{-4}$ and $y=3$ ⓑ $x=\mathrm{-2}$ and $x=\mathrm{-5}.$

Endeavor It three.43

Use slopes and y-intercepts to determine if the lines are parallel:

ⓐ $y=\mathrm{eight}$ and $y=\mathrm{-half-dozen}$ ⓑ $x=1$ and $x=\mathrm{-v}.$

Attempt It iii.44

Use slopes and y-intercepts to make up one's mind if the lines are parallel:

ⓐ $y=1$ and $y=\mathrm{-5}$ ⓑ $x=\mathrm{eight}$ and $x=\mathrm{-half-dozen}.$

Allow's wait at the lines whose equations are $y=\frac{\mathrm{i}}{4}x-1$ and $y=\mathrm{-4}x+\mathrm{ii},$ shown in Figure 3.9.

Figure 3.ix

These lines prevarication in the aforementioned aeroplane and intersect in right angles. We call these lines perpendicular.

If we wait at the slope of the starting time line, $m_1=\frac{\mathrm{i}}{4},$ and the slope of the 2nd line, ${\mathrm{grand}}_2=\mathrm{-4},$ we can run into that they are negative reciprocals of each other. If nosotros multiply them, their product is $\mathrm{-1}.$

$$\begin{array}{c}{\mathrm{one\; thousand}}_1·{\mathrm{one\; thousand}}_2\hfill \\ \frac{1}{\mathrm{four}}\left(\mathrm{-4}\right)\hfill \\ \mathrm{-1}\hfill \end{array}$$

This is ever truthful for perpendicular lines and leads united states to this definition.

Perpendicular Lines

Perpendicular lines are lines in the same aeroplane that form a correct bending.

• If $m_1$ and $m_2$ are the slopes of two perpendicular lines, and then:
  • their slopes are negative reciprocals of each other, ${\mathrm{yard}}_{\mathrm{ane}}=-\frac{1}{m_2}.$
  • the product of their slopes is $\mathrm{-one}$, $m_1·{\mathrm{one\; thousand}}_2=\mathrm{-1}.$
• A vertical line and a horizontal line are e'er perpendicular to each other.

We were able to look at the slope–intercept form of linear equations and decide whether or non the lines were parallel. We can practice the aforementioned affair for perpendicular lines.

We find the slope–intercept form of the equation, and then come across if the slopes are opposite reciprocals. If the production of the slopes is $\mathrm{-i},$ the lines are perpendicular.

Example iii.23

Use slopes to determine if the lines are perpendicular:

ⓐ $y=\mathrm{-v}x-4$ and $\mathrm{10}-5y=5$ ⓑ $\mathrm{seven}\mathrm{10}+2y=3$ and $\mathrm{two}x+7y=5$

Try It 3.45

Employ slopes to determine if the lines are perpendicular:

ⓐ $y=\mathrm{-iii}\mathrm{10}+2$ and $x-3y=4$ ⓑ $5\mathrm{10}+4y=\mathrm{one}$ and $4x+5y=\mathrm{three}.$

Endeavor It 3.46

Utilize slopes to determine if the lines are perpendicular:

ⓐ $y=2x-\mathrm{v}$ and $x+\mathrm{ii}y=\mathrm{-6}$ ⓑ $2\mathrm{10}-9y=\mathrm{iii}$ and $9x-\mathrm{two}y=\mathrm{i}.$

Section iii.two Exercises

Practice Makes Perfect

Discover the Slope of a Line

In the following exercises, find the slope of each line shown.

74 .

76 .

78 .

80 .

In the following exercises, find the slope of each line.

In the following exercises, use the slope formula to find the slope of the line between each pair of points.

85 .

$\left(2,\mathrm{five}\right),\left(4,0\right)$

86 .

$\left(3,\mathrm{half\; dozen}\right),\left(\mathrm{eight},0\right)$

87 .

$\left(\mathrm{-iii},\mathrm{three}\right),\left(4,\mathrm{-5}\right)$

88 .

$\left(\mathrm{-2},\mathrm{iv}\right),\left(\mathrm{three},\mathrm{-1}\right)$

89 .

$\left(\mathrm{-1},\mathrm{-2}\right),\left(2,5\right)$

xc .

$\left(\mathrm{-2},\mathrm{-1}\right),\left(6,5\right)$

91 .

$\left(\mathrm{four},\mathrm{-5}\right),\left(1,\mathrm{-two}\right)$

92 .

$\left(3,\mathrm{-6}\right),\left(\mathrm{ii},\mathrm{-ii}\right)$

Graph a Line Given a Point and the Gradient

In the following exercises, graph each line with the given signal and slope.

93 .

$\left(\mathrm{ii},5\right);$ $\mathrm{1000}=-\frac{1}{\mathrm{iii}}$

94 .

$\left(\mathrm{i},4\right)$ ; $m=-\frac{1}{\mathrm{two}}$

95 .

$\left(\mathrm{-ane},\mathrm{-4}\right)$ ; $\mathrm{yard}=\frac{4}{\mathrm{iii}}$

96 .

$\left(\mathrm{-three},\mathrm{-v}\right)$ ; $m=\frac{3}{2}$

97 .

y-intercept 3; $\mathrm{yard}=-\frac{2}{\mathrm{v}}$

98 .

x-intercept $\mathrm{-2}$ ; $m=\frac{\mathrm{three}}{4}$

99 .

$\left(\mathrm{-4},2\right)$ ; $m=4$

100 .

$\left(1,5\right)$ ; $\mathrm{grand}=\mathrm{-3}$

Graph a Line Using Its Slope and Intercept

In the post-obit exercises, identify the slope and y-intercept of each line.

101 .

$y=\mathrm{-seven}\mathrm{ten}+3$

102 .

$y=4x-\mathrm{ten}$

103 .

$\mathrm{iii}x+y=5$

104 .

$4\mathrm{10}+y=8$

105 .

$6x+4y=12$

106 .

$8x+3y=12$

107 .

$5x-2y=6$

108 .

$7x-3y=\mathrm{ix}$

In the post-obit exercises, graph the line of each equation using its gradient and y-intercept.

109 .

$y=\mathrm{three}\mathrm{10}-\mathrm{ane}$

110 .

$y=2\mathrm{ten}-3$

111 .

$y=\text{−}x+3$

112 .

$y=\text{−}x-4$

113 .

$y=-\frac{\mathrm{ii}}{5}\mathrm{10}-3$

114 .

$y=-\frac{3}{5}x+2$

115 .

$3x-2y=4$

116 .

$3x-4y=8$

Choose the Most Convenient Method to Graph a Line

In the following exercises, determine the well-nigh convenient method to graph each line.

119 .

$y=\mathrm{-3}x+4$

120 .

$x-y=5$

121 .

$\mathrm{ten}-y=\mathrm{ane}$

122 .

$y=\frac{\mathrm{two}}{3}x-\mathrm{ane}$

123 .

$3\mathrm{ten}-2y=\mathrm{-12}$

124 .

$2x-\mathrm{five}y=\mathrm{-10}$

Graph and Translate Applications of Slope–Intercept

125 .

The equation $P=31+1.75\mathrm{west}$ models the relation between the amount of Tuyet's monthly water bill payment, P, in dollars, and the number of units of water, w, used.

ⓐ Find Tuyet's payment for a month when 0 units of h2o are used.

ⓑ Find Tuyet's payment for a month when 12 units of h2o are used.

ⓒ Translate the slope and P-intercept of the equation.

ⓓ Graph the equation.

126 .

The equation $P=28+2.54w$ models the relation between the amount of R and y's monthly water bill payment, P, in dollars, and the number of units of water, w, used.

ⓐ Detect the payment for a month when R and y used 0 units of h2o.

ⓑ Find the payment for a calendar month when R and y used 15 units of water.

ⓒ Interpret the slope and P-intercept of the equation.

ⓓ Graph the equation.

127 .

Bruce drives his auto for his chore. The equation $R=0.575m+42$ models the relation betwixt the amount in dollars, R, that he is reimbursed and the number of miles, m, he drives in 1 twenty-four hours.

ⓐ Find the amount Bruce is reimbursed on a 24-hour interval when he drives 0 miles.

ⓑ Observe the amount Bruce is reimbursed on a day when he drives 220 miles.

ⓒ Translate the gradient and R-intercept of the equation.

ⓓ Graph the equation.

128 .

Janelle is planning to hire a automobile while on holiday. The equation $C=0.32m+15$ models the relation between the price in dollars, C, per day and the number of miles, thousand, she drives in one twenty-four hours.

ⓐ Find the cost if Janelle drives the auto 0 miles one day.

ⓑ Find the cost on a twenty-four hour period when Janelle drives the machine 400 miles.

ⓒ Interpret the slope and C-intercept of the equation.

ⓓ Graph the equation.

129 .

Cherie works in retail and her weekly salary includes commission for the amount she sells. The equation $\mathrm{Southward}=400+0.15c$ models the relation between her weekly salary, S, in dollars and the amount of her sales, c, in dollars.

ⓐ Observe Cherie's salary for a week when her sales were $0.

ⓑ Find Cherie'south salary for a week when her sales were $3,600.

ⓒ Interpret the slope and S-intercept of the equation.

ⓓ Graph the equation.

130 .

Patel'southward weekly salary includes a base pay plus committee on his sales. The equation $S=750+0.09c$ models the relation betwixt his weekly salary, S, in dollars and the amount of his sales, c, in dollars.

ⓐ Find Patel's salary for a calendar week when his sales were 0.

ⓑ Find Patel's salary for a calendar week when his sales were 18,540.

ⓒ Interpret the gradient and South-intercept of the equation.

ⓓ Graph the equation.

131 .

Costa is planning a dejeuner banquet. The equation $C=450+28g$ models the relation between the price in dollars, C, of the banquet and the number of guests, chiliad.

ⓐ Find the toll if the number of guests is 40.

ⓑ Find the price if the number of guests is 80.

ⓒ Interpret the slope and C-intercept of the equation.

ⓓ Graph the equation.

132 .

Margie is planning a dinner feast. The equation $C=750+42\mathrm{1000}$ models the relation between the cost in dollars, C of the banquet and the number of guests, g.

ⓐ Detect the price if the number of guests is 50.

ⓑ Find the cost if the number of guests is 100.

ⓒ Translate the gradient and C-intercept of the equation.

ⓓ Graph the equation.

Employ Slopes to Identify Parallel and Perpendicular Lines

In the following exercises, use slopes and y-intercepts to determine if the lines are parallel, perpendicular, or neither.

133 .

$y=\frac{\mathrm{three}}{4}\mathrm{10}-3;3\mathrm{ten}-\mathrm{four}y=\mathrm{-2}$

134 .

$3x+4y=-2;y=\frac{\mathrm{iii}}{\mathrm{iv}}x-3$

135 .

$\mathrm{ii}x-4y=\mathrm{six};\mathrm{ten}-2y=3$

136 .

$8x+6y=\mathrm{half\; dozen};12x+9y=12$

137 .

$x=5;x=\mathrm{-half\; dozen}$

138 .

$\mathrm{10}=-3;x=-2$

139 .

$4x-\mathrm{ii}y=5;\mathrm{three}x+\mathrm{half-dozen}y=\mathrm{viii}$

140 .

$8x-\mathrm{two}y=7;3x+12y=9$

141 .

$\mathrm{iii}x-6y=12;6\mathrm{10}-\mathrm{iii}y=3$

142 .

$9x-5y=\mathrm{four};5\mathrm{10}+9y=-1$

143 .

$7x-\mathrm{iv}y=\mathrm{viii};4x+7y=14$

144 .

$\mathrm{v}x-\mathrm{two}y=\mathrm{xi};\mathrm{five}x-y=7$

145 .

$3x-2y=\mathrm{viii};2\mathrm{ten}+3y=6$

146 .

$2x+3y=\mathrm{five};3\mathrm{ten}-\mathrm{two}y=7$

147 .

$3x-\mathrm{ii}y=1;2x-3y=\mathrm{ii}$

148 .

$2\mathrm{ten}+\mathrm{four}y=3;6x+3y=2$

149 .

$y=\mathrm{ii};y=\mathrm{half\; dozen}$

150 .

$y=-1;y=\mathrm{two}$

Writing Exercises

151 .

How does the graph of a line with gradient $m=\frac{\mathrm{ane}}{2}$ differ from the graph of a line with slope $\mathrm{thousand}=2?$

152 .

Why is the slope of a vertical line "undefined"?

153 .

Explain how you can graph a line given a point and its slope.

154 .

Explain in your ain words how to decide which method to utilise to graph a line.

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ Afterwards reviewing this checklist, what will yous practise to become confident for all objectives?

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Source: https://openstax.org/books/intermediate-algebra-2e/pages/3-2-slope-of-a-line

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