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y = −3

horizontal line

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y = 3 4 x 1

plotting points

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Graph and Interpret Applications of Slope–Intercept

Katherine is a private chef. The equation C = 6.5 m + 42 models the relation between her weekly cost, C , in dollars and the number of meals, m , that she serves.

  1. Find Katherine’s cost for a week when she serves no meals.
  2. Find the cost for a week when she serves 14 meals.
  3. Interpret the slope and C -intercept of the equation.
  4. Graph the equation.
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Marjorie teaches piano. The equation P = 35 h 250 models the relation between her weekly profit, P , in dollars and the number of student lessons, s , that she teaches.

  1. Find Marjorie’s profit for a week when she teaches no student lessons.
  2. Find the profit for a week when she teaches 20 student lessons.
  3. Interpret the slope and P –intercept of the equation.
  4. Graph the equation.

−$250  $450  The slope, 35, means that Marjorie’s weekly profit, P , increases by $35 for each additional student lesson she teaches. The P –intercept means that when the number of lessons is 0, Marjorie loses $250. 
The graph shows the x y-coordinate plane where h is plotted along the x-axis and P is potted along the y-axis. The x-axis runs from 0 to 24. The y-axis runs from negative 300 to 500. The line P equals 35 h minus 250 is plotted from the bottom left to the top right.

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Use Slopes to Identify Parallel Lines

In the following exercises, use slopes and y-intercepts to determine if the lines are parallel.

4 x 3 y = −1 ; y = 4 3 x 3

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2 x y = 8 ; x 2 y = 4

not parallel

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Use Slopes to Identify Perpendicular Lines

In the following exercises, use slopes and y-intercepts to determine if the lines are perpendicular.

y = 5 x 1 ; 10 x + 2 y = 0

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3 x 2 y = 5 ; 2 x + 3 y = 6

perpendicular

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Find the Equation of a Line

Find an Equation of the Line Given the Slope and y -Intercept

In the following exercises, find the equation of a line with given slope and y-intercept. Write the equation in slope–intercept form.

slope 1 3 and y -intercept ( 0 , −6 )

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slope −5 and y -intercept ( 0 , −3 )

y = −5 x 3

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slope 0 and y -intercept ( 0 , 4 )

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slope −2 and y -intercept ( 0 , 0 )

y = −2 x

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In the following exercises, find the equation of the line shown in each graph. Write the equation in slope–intercept form.

Find an Equation of the Line Given the Slope and a Point

In the following exercises, find the equation of a line with given slope and containing the given point. Write the equation in slope–intercept form.

m = 1 4 , point ( −8 , 3 )

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m = 3 5 , point ( 10 , 6 )

y = 3 5 x

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Horizontal line containing ( −2 , 7 )

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m = −2 , point ( −1 , −3 )

y = −2 x 5

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Find an Equation of the Line Given Two Points

In the following exercises, find the equation of a line containing the given points. Write the equation in slope–intercept form.

( 2 , 10 ) and ( −2 , −2 )

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( 7 , 1 ) and ( 5 , 0 )

y = 1 2 x 5 2

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( 3 , 8 ) and ( 3 , −4 ) .

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( 5 , 2 ) and ( −1 , 2 )

y = 2

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Find an Equation of a Line Parallel to a Given Line

In the following exercises, find an equation of a line parallel to the given line and contains the given point. Write the equation in slope–intercept form.

line y = −3 x + 6 , point ( 1 , −5 )

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line 2 x + 5 y = −10 , point ( 10 , 4 )

y = 2 5 x + 8

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line x = 4 , point ( −2 , −1 )

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line y = −5 , point ( −4 , 3 )

y = 3

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Find an Equation of a Line Perpendicular to a Given Line

In the following exercises, find an equation of a line perpendicular to the given line and contains the given point. Write the equation in slope–intercept form.

line y = 4 5 x + 2 , point ( 8 , 9 )

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line 2 x 3 y = 9 , point ( −4 , 0 )

y = 3 2 x 6

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line y = 3 , point ( −1 , −3 )

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line x = −5 point ( 2 , 1 )

y = 1

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Graph Linear Inequalities

Verify Solutions to an Inequality in Two Variables

In the following exercises, determine whether each ordered pair is a solution to the given inequality.

Determine whether each ordered pair is a solution to the inequality y < x 3 :

( 0 , 1 )
( −2 , −4 )
( 5 , 2 )
( 3 , −1 )
( −1 , −5 )

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Determine whether each ordered pair is a solution to the inequality x + y > 4 :

( 6 , 1 )
( −3 , 6 )
( 3 , 2 )
( −5 , 10 )
( 0 , 0 )

yes  no  yes  yes  no

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Recognize the Relation Between the Solutions of an Inequality and its Graph

In the following exercises, write the inequality shown by the shaded region.

Write the inequality shown by the graph with the boundary line y = x + 2 .

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals negative x plus 2 is plotted as a solid line extending from the top left toward the bottom right. The region below the line is shaded.
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Write the inequality shown by the graph with the boundary line y = 2 3 x 3 .

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals two-thirds x minus 3 is plotted as a dashed line extending from the bottom left toward the top right. The region above the line is shaded.

y > 2 3 x 3

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Write the inequality shown by the shaded region in the graph with the boundary line x + y = −4 .

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line x plus y equals negative 4 is plotted as a dashed line extending from the top left toward the bottom right. The region above the line is shaded.
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Write the inequality shown by the shaded region in the graph with the boundary line x 2 y = 6 .

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line x minus 2 y equals 6 is plotted as a solid line extending from the bottom left toward the top right. The region below the line is shaded.

x 2 y 6

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Graph Linear Inequalities

In the following exercises, graph each linear inequality.

Graph the linear inequality y > 2 5 x 4 .

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Graph the linear inequality y 1 4 x + 3 .

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals negative one-fourth x plus 3 is plotted as a solid line extending from the top left toward the bottom right. The region below the line is shaded.

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Graph the linear inequality x y 5 .

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Graph the linear inequality 3 x + 2 y > 10 .

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line 3 x plus 2 y equals 10 is plotted as a dashed line extending from the top left toward the bottom right. The region above the line is shaded.

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Graph the linear inequality y −3 x .

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Graph the linear inequality y < 6 .

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals 6 is plotted as a dashed, horizontal line. The region below the line is shaded.

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Practice test

Plot each point in a rectangular coordinate system.

( 2 , 5 )
( −1 , −3 )
( 0 , 2 )
( −4 , 3 2 )
( 5 , 0 )

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Which of the given ordered pairs are solutions to the equation 3 x y = 6 ?

( 3 , 3 )
( 2 , 0 )
( 4 , −6 )

yes  yes  no

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Find three solutions to the linear equation y = −2 x 4 .

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Find the x - and y -intercepts of the equation 4 x 3 y = 12 .

( 3 , 0 ) , ( 0 , −4 )

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Find the slope of each line shown.

Find the slope of the line between the points ( 5 , 2 ) and ( −1 , −4 ) .

1

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Graph the line with slope 1 2 containing the point ( −3 , −4 ) .

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Graph the line for each of the following equations.

Find the equation of each line. Write the equation in slope–intercept form.

slope 3 4 and y -intercept ( 0 , −2 )

y = 3 4 x 2

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m = 2 , point ( −3 , −1 )

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containing ( 10 , 1 ) and ( 6 , −1 )

y = 1 2 x 4

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parallel to the line y = 2 3 x 1 , containing the point ( −3 , 8 )

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perpendicular to the line y = 5 4 x + 2 , containing the point ( −10 , 3 )

y = 4 5 x 5

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Write the inequality shown by the graph with the boundary line y = x 3 .

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals negative x minus 3 is plotted. The solid line passes through the points (negative 3, 0) and (0, negative 3).
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Graph each linear inequality.

Questions & Answers

A veterinarian is enclosing a rectangular outdoor running area against his building for the dogs he cares for. He needs to maximize the area using 100 feet of fencing. The quadratic equation A=x(100−2x) gives the area, A , of the dog run for the length, x , of the building that will border the dog run. Find the length of the building that should border the dog run to give the maximum area, and then find the maximum area of the dog run.
Elizabeth Reply
ggfcc
Mike
Washing his dad’s car alone, eight year old Levi takes 2.5 hours. If his dad helps him, then it takes 1 hour. How long does it take the Levi’s dad to wash the car by himself?
Gagan Reply
1,75hrs
Mike
I'm going to guess. Divide Levi's time by 2. Then divide 1 hour by 2. 1.25 + 0.5 = 1.3?
John
Oops I mean 1.75
John
I'm guessing this because since I have divide 1 hour by 2, I have to do the same for the 2.5 hours it takes Levi by himself.
John
Drew burned 1,800 calories Friday playing 1 hour of basketball and canoeing for 2 hours. On Saturday, he spent 2 hours playing basketball and 3 hours canoeing and burned 3,200 calories. How many calories did he burn per hour when playing basketball?
Marie Reply
Brandon has a cup of quarters and dimes with a total value of $3.80. The number of quarters is four less than twice the number of dimes. How many quarters and how many dimes does Brandon have?
Kendra Reply
Tickets to a Broadway show cost $35 for adults and $15 for children. The total receipts for 1650 tickets at one performance were $47,150. How many adult and how many child tickets were sold?
dana Reply
825
Carol
Arnold invested $64,000, some at 5.5% interest and the rest at 9%. How much did he invest at each rate if he received $4,500 in interest in one year? How do I do this
Tanesia Reply
how to square
Fiona Reply
easiest way to find the square root of a large number?
Jackie
the accompanying figure shows known flow rates of hydrocarbons into and out of a network of pipes at an oil refinery set up a linear system whose solution provides the unknown flow rates (b) solve the system for the unknown flow rates (c) find the flow rates and directions of flow if x4=50and x6=0
Sabee Reply
What is observation
adeyemi Reply
I'm confused by the question. Can you describe or explain the math question it pertains to?
Melissa
there is no math to it because all you use is your vision or gaze to the sorrounding areas
Cesarp
Teegan likes to play golf. He has budgeted $60 next month for the driving range. It costs him $10.55 for a bucket of balls each time he goes. What is the maximum number of times he can go to the driving range next month?
Sunnyshay Reply
5 times max
Anton
Felecia left her home to visit her daughter, driving 45mph. Her husband waited for the dog sitter to arrive and left home 20 minutes, or 1/3 hour later. He drove 55mph to catch up to Felecia. How long before he reaches her?
Sophia Reply
35 min
Debra
Carmen wants to tile the floor of his house. He will need 1,000 square feet of tile. He will do most of the floor with a tile that costs $1.50 per square foot, but also wants to use an accent tile that costs $9.00 per square foot. How many square feet of each tile should he plan to use if he wants the overall cost to be $3 per square foot?
Parker Reply
what you wanna get
Cesar
800 sq. ft @ $1.50 & 200 sq. ft @ $9.00
Marco
Geneva treated her parents to dinner at their favorite restaurant. The bill was $74.25. Geneva wants to leave 16 % of the total - bill as a tip. How much should the tip be?
74.25 × .16 then get the total and that will be your tip
David
$74.25 x 0.16 = $11.88 total bill: $74.25 + $11.88 = $86.13
ericka
yes and tip 16% will be $11.88
David
what is the shorter way to do it
Cesar Reply
Priam has dimes and pennies in a cup holder in his car. The total value of the coins is $4.21. The number of dimes is three less than four times the number of pennies. How many dimes and how many pennies are in the cup?
Alexandra Reply
Practice Key Terms 3

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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