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 = (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
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, (x_{1}, y_{1}) and (x_{2}, y_{2}), to find the slope.
Find the ratio between change in y-values and change in x.
m = (y_{2} - y_{1}) / (x_{2} - x_{1})
Substitute :
(x_{1}, y_{1}) = (0, 8)
(x_{2}, y_{2}) = (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, (x₁, y₁) and (x₂, y₂), to find the slope.
Find the ratio between change in y-values and change in x.
m = (y_{2} - y_{1})/(x_{2} - x_{1})
Substitute :
(x_{1}, y_{1}) = (0, 25)
(x_{2}, y_{2}) = (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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