We can use information from a description of a linear relationship to find the slope and y-intercept and to write an equation.
Example 1 :
The rent charged for space in an office building is a linear relationship related to the size of the space rented. At west main street office rentals, $750 rent charged for 600 square feet of space and $1150 rent charged for 900 square feet of space. Write an equation in slope-intercept form for the rent at West Main Street Office Rentals.
Solution :
Step 1 :
Identify the independent and dependent variables.
The independent variable (x) is the square footage of floor space.
The dependent variable (y) is the monthly rent.
Step 2 :
Write the information given in the problem as ordered pairs.
The rent for 600 square feet of floor space is $750 :
(600, 750)
The rent for 900 square feet of floor space is $1150 :
(900, 1150)
Step 3 :
Find the slope.
m = (y_{2} - y_{1}) / (x_{2} - x_{1})
Substitute :
(x_{1}, y_{1}) = (600, 750)
(x_{2}, y_{2}) = (900, 1150)
Then,
m = (1150 - 750) / (900 - 600)
m = 400 / 300
m = 4/3
Step 4 :
Find the y-intercept.
Use the slope 4/3 and one of the ordered pairs (600, 750).
Slope-intercept form :
y = mx + b
Substitute m = 4/3, x = 600 and y = 750.
750 = (4/3)(600) + b
750 = (4)(200) + b
750 = 800 + b
-50 = b
Step 5 :
Substitute the slope and y-intercept.
Slope-intercept form
y = mx + b
Substitute m = 4/3 and b = -50
y = (4/3)x + (-50)
y = (4/3)x - 50
Example 2 :
Hari’s weekly allowance varies depending on the number of chores he does. He received $16 in allowance the week he did 12 chores, and $14 in allowance the week he did 8 chores. Write an equation for his allowance in slope-intercept form.
Solution :
Step 1 :
Identify the independent and dependent variables.
The independent variable (x) is number of chores Hari does per week
The dependent variable (y) is the allowance he receives per week.
Step 2 :
Write the information given in the problem as ordered pairs.
For 12 chores, he receives $16 allowance :
(12, 16)
For 8 chores, he receives $14 allowance :
(8, 14)
Step 3 :
Find the slope.
m = (y_{2} - y_{1}) / (x_{2} - x_{1})
Substitute :
(x_{1}, y_{1}) = (12, 16)
(x_{2}, y_{2}) = (8, 14)
Then,
m = (14 - 16) / (8 - 12)
m = (-2) / (-4)
m = 1/2
m = 0.5
Step 4 :
Find the y-intercept.
Use the slope 0.5 and one of the ordered pairs (8, 14).
Slope-intercept form :
y = mx + b
Substitute m = 0.5, x = 8 and y = 14.
14 = (0.5)(8) + b
14 = 4 + b
10 = b
Step 5 :
Substitute the slope and y-intercept.
Slope-intercept form
y = mx + b
Substitute m = 0.5 and b = 10
y = 0.5x + 10
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