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4.5 Use the slope–intercept form of an equation of a line  (Page 9/12)

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The equation P = 31 + 1.75 w 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.

  1. ⓐ Find Tuyet’s payment for a month when 0 units of water are used.
  2. ⓑ Find Tuyet’s payment for a month when 12 units of water are used.
  3. ⓒ Interpret the slope and P -intercept of the equation.
  4. ⓓ Graph the equation.
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The equation P = 28 + 2.54 w models the relation between the amount of Randy’s monthly water bill payment, P , in dollars, and the number of units of water, w , used.

  1. ⓐ Find the payment for a month when Randy used 0 units of water.
  2. ⓑ Find the payment for a month when Randy used 15 units of water.
  3. ⓒ Interpret the slope and P -intercept of the equation.
  4. ⓓ Graph the equation.
  1. ⓐ $28
  2. ⓑ $66.10
  3. ⓒ The slope, 2.54, means that Randy’s payment, P , increases by $2.54 when the number of units of water he used, w, increases by 1. The P –intercept means that if the number units of water Randy used was 0, the payment would be $28.
  4. ⓓ

    The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane represents the variable w and runs from negative 2 to 20. The y-axis of the plane represents the variable P and runs from negative 1 to 100. The line begins at the point (0, 28) and goes through the point (15, 66.1).

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Bruce drives his car for his job. The equation R = 0.575 m + 42 models the relation between the amount in dollars, R , that he is reimbursed and the number of miles, m , he drives in one day.

  1. ⓐ Find the amount Bruce is reimbursed on a day when he drives 0 miles.
  2. ⓑ Find the amount Bruce is reimbursed on a day when he drives 220 miles.
  3. ⓒ Interpret the slope and R -intercept of the equation.
  4. ⓓ Graph the equation.
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Janelle is planning to rent a car while on vacation. The equation C = 0.32 m + 15 models the relation between the cost in dollars, C , per day and the number of miles, m , she drives in one day.

  1. ⓐ Find the cost if Janelle drives the car 0 miles one day.
  2. ⓑ Find the cost on a day when Janelle drives the car 400 miles.
  3. ⓒ Interpret the slope and C –intercept of the equation.
  4. ⓓ Graph the equation.
  1. ⓐ $15
  2. ⓑ $143
  3. ⓒ The slope, 0.32, means that the cost, C , increases by $0.32 when the number of miles driven, m, increases by 1. The C -intercept means that if Janelle drives 0 miles one day, the cost would be $15.
  4. ⓓ

    The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane represents the variable m and runs from negative 1 to 500. The y-axis of the plane represents the variable C and runs from negative 1 to 200. The line begins at the point (0,15) and goes through the point (400,143).

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

  1. ⓐ Find Cherie’s salary for a week when her sales were 0.
  2. ⓑ Find Cherie’s salary for a week when her sales were 3600.
  3. ⓒ Interpret the slope and S –intercept of the equation.
  4. ⓓ Graph the equation.
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Patel’s weekly salary includes a base pay plus commission on his sales. The equation S = 750 + 0.09 c models the relation between his weekly salary, S , in dollars and the amount of his sales, c , in dollars.

  1. ⓐ Find Patel’s salary for a week when his sales were 0.
  2. ⓑ Find Patel’s salary for a week when his sales were 18,540.
  3. ⓒ Interpret the slope and S -intercept of the equation.
  4. ⓓ Graph the equation.
  1. ⓐ $750
  2. ⓑ $2418.60
  3. ⓒ The slope, 0.09, means that Patel’s salary, S , increases by $0.09 for every $1 increase in his sales. The S -intercept means that when his sales are $0, his salary is $750.
  4. ⓓ

    The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane represents the variable w and runs from negative 1 to 20000. The y-axis of the plane represents the variable P and runs from negative 1 to 3000. The line begins at the point (0, 750) and goes through the point (18540, 2415).

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

  1. ⓐ Find the cost if the number of guests is 40.
  2. ⓑ Find the cost if the number of guests is 80.
  3. ⓒ Interpret the slope and C -intercept of the equation.
  4. ⓓ Graph the equation.
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Margie is planning a dinner banquet. The equation C = 750 + 42 g models the relation between the cost in dollars, C of the banquet and the number of guests, g .

  1. ⓐ Find the cost if the number of guests is 50.
  2. ⓑ Find the cost if the number of guests is 100.
  3. ⓒ Interpret the slope and C –intercept of the equation.
  4. ⓓ Graph the equation.
  1. ⓐ $2850
  2. ⓑ $4950
  3. ⓒ The slope, 42, means that the cost, C , increases by $42 for when the number of guests increases by 1. The C -intercept means that when the number of guests is 0, the cost would be $750.
  4. ⓓ

    The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane represents the variable g and runs from negative 1 to 150. The y-axis of the plane represents the variable C and runs from negative 1 to 7000. The line begins at the point (0, 750) and goes through the point (100, 4950).

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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.

y = 3 4 x − 3 ; 3 x − 4 y = − 2

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y = 2 3 x − 1 ; 2 x − 3 y = − 2

parallel

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2 x − 5 y = − 3 ; y = 2 5 x + 1

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3 x − 4 y = − 2 ; y = 3 4 x − 3

parallel

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2 x − 4 y = 6 ; x − 2 y = 3

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6 x − 3 y = 9 ; 2 x − y = 3

not parallel

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

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8 x + 6 y = 6 ; 12 x + 9 y = 12

parallel

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x = 5 ; x = − 6

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x = 7 ; x = − 8

parallel

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x = − 4 ; x = − 1

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x = − 3 ; x = − 2

parallel

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y = 2 ; y = 6

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y = 5 ; y = 1

parallel

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y = − 4 ; y = 3

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y = − 1 ; y = 2

parallel

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x − y = 2 ; 2 x − 2 y = 4

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4 x + 4 y = 8 ; x + y = 2

not parallel

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x − 3 y = 6 ; 2 x − 6 y = 12

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5 x − 2 y = 11 ; 5 x − y = 7

not parallel

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3 x − 6 y = 12 ; 6 x − 3 y = 3

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4 x − 8 y = 16 ; x − 2 y = 4

not parallel

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9 x − 3 y = 6 ; 3 x − y = 2

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x − 5 y = 10 ; 5 x − y = − 10

not parallel

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7 x − 4 y = 8 ; 4 x + 7 y = 14

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9 x − 5 y = 4 ; 5 x + 9 y = − 1

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.

3 x − 2 y = 8 ; 2 x + 3 y = 6

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x − 4 y = 8 ; 4 x + y = 2

perpendicular

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2 x + 5 y = 3 ; 5 x − 2 y = 6

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2 x + 3 y = 5 ; 3 x − 2 y = 7

perpendicular

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3 x − 2 y = 1 ; 2 x − 3 y = 2

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3 x − 4 y = 8 ; 4 x − 3 y = 6

not perpendicular

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5 x + 2 y = 6 ; 2 x + 5 y = 8

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2 x + 4 y = 3 ; 6 x + 3 y = 2

not perpendicular

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4 x − 2 y = 5 ; 3 x + 6 y = 8

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2 x − 6 y = 4 ; 12 x + 4 y = 9

perpendicular

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6 x − 4 y = 5 ; 8 x + 12 y = 3

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8 x − 2 y = 7 ; 3 x + 12 y = 9

perpendicular

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Everyday math

The equation C = 5 9 F − 17.8 can be used to convert temperatures F , on the Fahrenheit scale to temperatures, C , on the Celsius scale.

  1. ⓐ Explain what the slope of the equation means.
  2. ⓑ Explain what the C –intercept of the equation means.
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The equation n = 4 T − 160 is used to estimate the number of cricket chirps, n , in one minute based on the temperature in degrees Fahrenheit, T .

  1. ⓐ Explain what the slope of the equation means.
  2. ⓑ Explain what the n –intercept of the equation means. Is this a realistic situation?
  1. ⓐ For every increase of one degree Fahrenheit, the number of chirps increases by four.
  2. ⓑ There would be −160 chirps when the Fahrenheit temperature is 0 ° . (Notice that this does not make sense; this model cannot be used for all possible temperatures.)
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Writing exercises

Explain in your own words how to decide which method to use to graph a line.

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Why are all horizontal lines parallel?

Answers will vary.

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Self check

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

This table has eight rows and four columns. The first row is a header row and it labels each column. The first column is labeled

ⓑ After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?

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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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