Two yards of fabric costs $13 and 5 yards of fabric costs $32.50.
Write an equation in slope-intercept form to describe the cost, y, for each yard x
To determine the equation in the slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept, we first need to find the slope (m).
The slope can be found using the formula:
m = (change in y) / (change in x)
Given two points: (x1, y1) = (2, $13) and (x2, y2) = (5, $32.50), we can find the slope as follows:
m = (y2 - y1) / (x2 - x1)
m = ($32.50 - $13) / (5 - 2)
m = $19.50 / 3
m = $6.50
This slope $6.50 represents the cost per yard of fabric.
Now we use one of the given points to solve for the y-intercept, b. We can use the point (2, $13):
y = mx + b
$13 = $6.50(2) + b
$13 = $13 + b
$13 - $13 = b
b = $0
This means that the y-intercept, which represents the starting cost at 0 yards, is $0. This is consistent with what we generally expect for buying fabric: if you buy 0 yards, it should cost $0.
Now we can write the equation in slope-intercept form:
y = $6.50x + $0
Or simply:
y = $6.50x
This is the equation that describes the cost (y) for each yard (x) of fabric.