The slope-intercept form is one of the ways to write a linear equation (the equation of a line). The slope-intercept form is written as y = mx + b. Here m stands for slope and b stands for y-intercept (the point at where the line cuts y-axis).
We can model many real-world problems as slope intercept form equation and solve them easily.
Example 1 :
Greta makes clay mugs and bowls as gifts at the Crafty Studio. She pays a membership fee of $15 a month and an equipment fee of $3.00 an hour to use the potter’s wheel, table, and kiln. Write an equation in the form y = mx + b that Greta can use to calculate her monthly costs.
Solution :
Step 1 :
Find the independent variable x, for the given situation.
The number of hours Greta uses the studio
Step 2 :
Find the dependent variable y, for the given situation.
The money Greta pays the studio each month
Step 3 :
For example, during April, Greta does not use the equipment at all.
Find the number of hours (x) for April.
0
Find her cost (y) for April.
$15
Find the y-intercept "b" in the equation.
15
Step 4 :
For example, during May, Greta uses the equipment for 8 hours.
Find her cost (y) for May.
$15 + 8($3) = $39
In June, if she spends 11 hours, then her cost is $48.
From May to June, the change in x-values is +3.
From May to June, the change in y-values is +9.
Find the slope "m" in the equation y = mx + b.
m = change in y-values/change in x-values
m = 9/3
m = 3
Step 5 :
Use the values for m and b to write an equation for Greta’s costs in the form y = mx + b :
y = 3x + 15
Example 2 :
Ken has a weekly goal of burning 2400 calories by taking brisk walks. Ken would like to burn 300 calories per hour. Write an equation in the form y = mx + b that can use to calculate number of calories left to burn.
Solution :
Step 1 :
Find the independent variable x, for the given situation.
Number of hours taken by Ken to take brisk walk.
Step 2 :
Find the dependent variable y, for the given situation.
Number of calories left to burn after "x" hours.
Step 3 :
For example, on the first day of the week, Ken does take brisk walk.
Since, he does not take brisk walk on the first day of the week, the value of "x" is
0
Number of calories left to burn after the first day :
2400
Find the y-intercept "b" in the equation.
2400
Step 4 :
For example, Ken takes brisk walks for 1 hour on the second day of the week.
Find the number of calories left to burn after the second day.
2400 - 300(1) = 2100
On the third day, if he take brisk walks for 2 hours, then the number of calories left to burn after the third day is
1500
From the first day to third day, the change in x-values is
+3.
From the first day to third day, the change in y-values is
-900.
(Since the calories burned, negative sign is taken)
Find the slope "m" in the equation y = mx + b.
m = change in y-values/change in x-values
m = -900/3
m = -300
Step 5 :
Use the values for m and b to write an equation for Greta’s costs in the form y = mx + b :
y = -300x + 2400
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