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.
We can find the slope of the line that describes the cost function by dividing the change in cost by the change in yards. Since we have two points representing two separate purchases:
Point 1 (x1, y1) = (2 yards, $13)
Point 2 (x2, y2) = (5 yards, $32.50)
We can calculate the slope (m) as follows:
m = (y2 - y1) / (x2 - x1)
m = ($32.50 - $13) / (5 yards - 2 yards)
m = $19.50 / 3 yards
m = $6.50 per yard
The slope (m), which is the rate of change, tells us that each yard of fabric costs $6.50.
Now, let's find the y-intercept (b), which is the cost when x = 0. Since we don't have a direct value for when x = 0, we can use the slope and one of the points provided to solve for b. Let's use point 1 for this:
y1 = mx1 + b
$13 = $6.50(2) + b
$13 = $13 + b
$13 - $13 = b
b = $0
Here, the y-intercept (b) seems to be zero. However, typically that would not make much sense in the context of fabric, since there is usually a minimum cut fee or base cost even for a small quantity of fabric less than a yard. In this particular case, either the given data points imply there is no cost for zero yards (which is unlikely in a real-world scenario) or the given transactions include some fixed cost that gets averaged out over the yards purchased and it's not visible to us.
That being said, the equation in slope-intercept form (y = mx + b) for the fabric cost based on the given information would be:
y = $6.50x + $0
Which simplifies to:
y = $6.50x
This equation states that the cost (y) for any number of yards (x) is equal to $6.50 times the number of yards.