The slope or rate of change and the y-intercept or initial value for each of the following situations is as follows:
1) Sarah's Heavenly Cupcakes: Slope = 1.5; y-intercept = 0
2) Frank's Pizzeria: Slope = 1.75; y-intercept = 10
3) Marco's Computer Repairs: Slope = 15; y-intercept = 12.99
4) Linda's Dog Walking Service: Slope = 0.25; y-intercept = 10
5) Phil's Frozen Yogurt Shop: Slope = 0.5; y-intercept = 3
6) Ivy's Lawn Care Service: Slope = 0.5; y-intercept = 30
7) AI's Hot Dog Snacks: Slope = x is 5, y = 2, y-intercepts = 0
8) Carol's Thrift Shop: Slope = 4.99; y-intercept = 0
9) Wilma's Drop-off Child Care Services: Slope = 7.5; y-intercept = 0.
The slope is the gradient or the rate of change. It is computed using the linear equation y = mx + b, where m is the slope, x is a constant, and b is the y-intercept.
The slope can also be computed using the direct formula, m = y2-y1/x2-x1, which represents the Rise/Run.
1) Sarah's Heavenly Cupcakes
Number of Cupcakes Cost
1 $1.50
2 $3.00
3 $4.50
4 $6.00
Slope, m = y2-y1/x2-x1
= (6 - 3)/(4 -2)
= 3/2
= 1.5
y-intercept, y = mx + b
= 6 = 1.5(4) + b
= 6 = 6 +b
b = 6 - 6 = 0
y = 0
2) Frank's Pizzeria:
y = 10 + 1.75x
Slope = 1.75
y-intercept = 10
3) Marco's Computer Repairs:
The total cost for 2 hours = $42.99
The total cost for 5 hours = $87.99
Slope, m = y2-y1/x2-x1
= (87.99 - 42.99)/(5 - 2)
= 45/3
= 15
y-intercept, y = mx + b
87.99 = 15(5) + b
87.99 = 75 + b
b = 12.99
4) Linda's Dog Walking Service:
Slope, m = y2-y1/x2-x1
= (20 - 10)/(40 - 0)
= 10/40
= 0.25
y-intercept, y = mx + b
20 = 0.25(40) + b
20 = 10 + b
b = 10
y = 10 + 0.25x
5) Phil's Frozen Yogurt Shop:
The cost of 5 oz of frozen yogurt = $3
Toppings = $0.50 each
Let the number of toppings = x
Let the total cost = y
y = 3 + 0.5x
6) Ivy's Lawn Care Service:
Number Total
of Minutes Cost
15 $37.50
20 $40.00
40 $50.00
55 $57.50
Slope, m = y2-y1/x2-x1
= (57.50 - 40)/(55 - 20)
= 17.5/35
= 0.5
y-intercept, y = mx + b
57.50 = 0.5(55) + b
57.50 = 27.50 + b
b = 30
7) AI's Hot Dog Snacks:
Total cost of a soda and 2 hot dogs = $7
Total cost of a soda and 4 hot dogs = $12
Let the cost per soda = x
Let the cost per hot dog = y
Equations:
Customer A: x + 2y = 7
Customer B: x + 4y = 12
Subtract Equation 1 from Equation 2:
x + 4y = 12
-
x + 2y = 7
2y = 5
y = 2.5
Substitute y = 2.5 in Equation 1 or 2:
x + 2(2.5) = 7
x = 5 = 7
x = 2
8) Carol's Thrift Shop:
y = 4.99x
Slope = 4.99
y-intercept = 0
9) Wilma's Drop-off Child Care Services:
Slope, m = y2-y1/x2-x1
= (30 - 15)/(4 - 2)
= 15/2
= 7.5
y-intercept, y = mx + b
30 = 7.5(4) + b
30 = 30 + b
b = 0