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4.5 Use the slope–intercept form of an equation of a line  (Page 5/12)

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The cost of running some types business has two components—a fixed cost and a variable cost . The fixed cost is always the same 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.

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

ⓐ Find Stella’s cost for a week when she sells no pizzas.
ⓑ Find the cost for a week when she sells 15 pizzas.
ⓒ Interpret the slope and C -intercept of the equation.
ⓓ Graph the equation.

Solution

ⓐ Find Stella's cost for a week when she sells no pizzas. .
Find C when p = 0 . .
Simplify. .
Stella's fixed cost is $25 when she sells no pizzas.
ⓑ Find the cost for a week when she sells 15 pizzas. .
Find C when p = 15 . .
Simplify. .
.
Stella's costs are $85 when she sells 15 pizzas.
ⓒ Interpret the slope and C -intercept of the equation. .
The slope, 4, means that the cost increases by $4 for each pizza Stella sells. The C -intercept means that even when Stella sells no pizzas, her costs for the week are $25.
ⓓ Graph the equation. We'll need to use a larger scale than our usual. Start at the C -intercept (0, 25) then count out the rise of 4 and the run of 1 to get a second point. .

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Sam drives a delivery van. The equation C = 0.5 m + 60 models the relation between his weekly cost, 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 cost for a week when he drives 250 miles.
ⓒ Interpret the slope and C -intercept of the equation.
ⓓ Graph the equation.

  1. ⓐ $60
  2. ⓑ $185
  3. ⓒ The slope, 0.5, means that the weekly cost, C , increases by $0.50 when the number of miles driven, n, increases by 1. The C -intercept means that when the number of miles driven is 0, the weekly cost is $60
  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 10 to 400. The y-axis of the plane represents the variable C and runs from negative 10 to 300. The line begins at the point (0, 65) and goes through the point (250, 185).

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

ⓐ Find Loreen’s cost for a week when she writes no invitations.
ⓑ Find the cost for a week when she writes 75 invitations.
ⓒ Interpret the slope and C -intercept of the equation.
ⓓ Graph the equation.

  1. ⓐ $35
  2. ⓑ $170
  3. ⓒ The slope, 1.8, means that the weekly cost, C, increases by $1.80 when the number of invitations, n , increases by 1.80.
    The C -intercept means that when the number of invitations is 0, the weekly cost is $35.;
  4. ⓓ

    The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane represents the variable n and runs from negative 10 to 400. The y-axis of the plane represents the variable C and runs from negative 10 to 300. The line begins at the point (0, 35) and goes through the point (75, 170).

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Use slopes to identify parallel lines

The slope of a line indicates how steep the line is and whether it rises or falls as we read it from left to right. Two lines that have the same slope are called parallel lines. Parallel lines never intersect.

The figure shows three pairs of lines side-by-side. The pair of lines on the left run diagonally rising from left to right. The pair run side-by-side, not crossing. The pair of lines in the middle run diagonally dropping from left to right. The pair run side-by-side, not crossing. The pair of lines on the right run diagonally also dropping from left to right, but with a lesser slope. The pair run side-by-side, not crossing.

We say this more formally in terms of the rectangular coordinate system. Two lines that have the same slope and different y -intercepts are called parallel lines    . See [link] .

The figure shows two lines graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 8 to 8. The y-axis of the plane runs from negative 8 to 8. One line goes through the points (negative 5,1) and (5,5). The other line goes through the points (negative 5, negative 4) and (5,0).
Verify that both lines have the same slope, m = 2 5 , and different y -intercepts.
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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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