WRITING AN EQUATION FROM A DESCRIPTION

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  =  (y2 - y1) / (x2 - x1)

Substitute : 

(x1, y1)  =  (600, 750)

(x2, y2)  =  (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  =  (y2 - y1) / (x2 - x1)

Substitute : 

(x1, y1)  =  (12, 16)

(x2, y2)  =  (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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