WRITING LINEAR EQUATIONS FROM SITUATIONS AND GRAPHS

In this section, we are going to see how to write linear equations from the given situations and graphs in slope-intercept form.

The slope-intercept form of the equation is y = mx + b, where "m" is slope and "b" is y-intercept. And y-intercept is nothing but the point at which the line cuts y-axis.

For example, if the line cuts y-axis at (0, -4), then the y-intercept is -4.

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

Example 3 :

A video club charges a one-time membership fee plus a rental fee for each DVD borrowed. Use the graph to write an equation in slope-intercept form to represent the amount spent, y, on x DVD rentals.

Solution :

Step 1 :

Choose two points on the graph, (x1, y1) and (x2, y2), to find the slope.

Find the ratio between change in y-values and change in x.

m  =  (y2 - y1) / (x2 - x1)

Substitute :

(x1, y1) = (0, 8)

(x2, y2) = (8, 18)

Then,

m = (18 - 8)/(8 - 0)

m = 10/8

m = 1.25

Step 2 :

Read the y-intercept from the graph. That is, the point at which the line cuts y-axis.

The y-intercept is 8.

b = 8

Step 3 :

Let us use slope (m) and y-intercept (b) values to write an equation in slope-intercept form.

y = mx + b (Slope-intercept form)

Substitute m = 1.25 and b = 8.

y = 1.25x + 8

Example 4 :

The cash register subtracts $2.50 from a $25 Coffee Café gift card for every medium coffee the customer buys. Use the graph to write an equation in slope-intercept form to represent this situation.

Solution :

Step 1 :

Choose two points on the graph, (x, y) and (x, y), to find the slope.

Find the ratio between change in y-values and change in x.

m = (y2 - y1)/(x2 - x1)

Substitute :

(x1, y1) = (0, 25)

(x2, y2) = (10, 0)

Then,

m = (0 - 25)/(10 - 0)

m = -25/10

m = -2.5

Step 2 :

Read the y-intercept from the graph. That is, the point at which the line cuts y-axis.

The y-intercept is 25.

Step 3 :

Let us use slope (m) and y-intercept (b) values to write an equation in slope-intercept form.

y = -2.5x + 25 (Slope-intercept form)

Substitute m = -2.5 and b = 25.

y = -2.5x + 25

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