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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.

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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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