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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

4x+7y=29,x+3y=11 substitute method of linear equation
Srinu Reply
substitute method of linear equation
Srinu
Solve one equation for one variable. Using the 2nd equation, x=11-3y. Substitute that for x in first equation. this will find y. then use the value for y to find the value for x.
bruce
I want to learn
Elizebeth
help
Elizebeth
I want to learn. Please teach me?
Wayne
1) Use any equation, and solve for any of the variables. Since the coefficient of x (the number in front of the x) in the second equation is 1 (it actually isn't shown, but 1 * x = x), use that equation. Subtract 3y from both sides (this isolates the x on the left side of the equal sign).
bruce
2) This results in x=11-3y. x is note in terms of y. Use that as the value of x and substitute for all x in the first equation. The first equation becomes 4(11-3y)+7y =29. Note that the only variable left in the first equation is the y. If you have multiple variable, then something is wrong.
bruce
3) Distribute (multiply) the 4 across 11-3y to get 44-12y. Add this to the 7y. So, the equation is now 44-5y=29.
bruce
4) Solve 44-5y=29 for y. Isolate the y by subtracting 44 from birth sides, resulting in -5y=-15. Now, divide birth sides by -5 (since you have -5y). This results in y=3. You now have the value of one variable.
bruce
5) The last step is to take the value of y from Step 4) and substitute into the 2nd equation. Therefore: x+3y=11 becomes x+3(3)=11. Then multiplying, x+9=11. Finally, solve for x by subtracting 9 from both sides. Therefore, x=2.
bruce
6) The ordered pair of (2, 3) is the proposed solution. To check, substitute those values into either equation. If the result is true, then the solution is correct. 4(2)+7(3)=8+21=29. TRUE! Finished.
bruce
At 1:30 Marlon left his house to go to the beach, a distance of 5.625 miles. He rose his skateboard until 2:15, and then walked the rest of the way. He arrived at the beach at 3:00. Marlon's speed on his skateboard is 1.5 times his walking speed. Find his speed when skateboarding and when walking.
Andrew Reply
divide 3x⁴-4x³-3x-1 by x-3
Ritik Reply
how to multiply the monomial
Ceny Reply
Two sisters like to compete on their bike rides. Tamara can go 4 mph faster than her sister, Samantha. If it takes Samantha 1 hours longer than Tamara to go 80 miles, how fast can Samantha ride her bike? Got questions? Get instant answers now!
Seera Reply
how do u solve that question
Seera
Two sisters like to compete on their bike rides. Tamara can go 4 mph faster than her sister, Samantha. If it takes Samantha 1 hours longer than Tamara to go 80 miles, how fast can Samantha ride her bike?
Seera
Speed=distance ÷ time
Tremayne
x-3y =1; 3x-2y+4=0 graph
Juned Reply
Brandon has a cup of quarters and dimes with a total of 5.55$. The number of quarters is five less than three times the number of dimes
ashley Reply
app is wrong how can 350 be divisible by 3.
Raheem Reply
June needs 48 gallons of punch for a party and has two different coolers to carry it in. The bigger cooler is five times as large as the smaller cooler. How many gallons can each cooler hold?
Susanna Reply
Susanna if the first cooler holds five times the gallons then the other cooler. The big cooler holda 40 gallons and the 2nd will hold 8 gallons is that correct?
Georgie
@Susanna that person is correct if you divide 40 by 8 you can see it's 5 it's simple
Ashley
@Geogie my bad that was meant for u
Ashley
Hi everyone, I'm glad to be connected with you all. from France.
Lorris Reply
I'm getting "math processing error" on math problems. Anyone know why?
Ray Reply
Can you all help me I don't get any of this
Jade Reply
4^×=9
Alberto Reply
Did anyone else have trouble getting in quiz link for linear inequalities?
Sireka Reply
operation of trinomial
Justin 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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