Last year, Alayah ordered 4 cubic yards of soil and spent $220. This year, she orders 7 cubic yards of soil and spends $340. Write the equation of the linear function that models the relationship between the number of cubic yards of soil ordered and the total cost in y = mx + b form.
First, we need to find the slope, or rate of change, of the function. To do this, we can use the formula:
slope = (change in y) / (change in x)
In this case, the change in y is the difference in cost between the two orders ($340 - $220 = $120), and the change in x is the difference in the number of cubic yards ordered (7 - 4 = 3). So:
slope = 120 / 3 = 40
This means that for every additional cubic yard of soil ordered, the total cost increases by $40.
Next, we need to find the y-intercept, or the value of y when x is zero. In other words, how much would it cost if Alayah didn't order any soil? We can use either of the two data points to find this. Let's use the first one:
y = mx + b
220 = 4(40) + b
220 = 160 + b
b = 60
So the equation in y = mx + b form is:
y = 40x + 60
This means that the total cost (y) is equal to 40 times the number of cubic yards ordered (x), plus a base cost of $60.
To write the equation in y = mx + b form, we need to find the values of m and b.
Let's consider the number of cubic yards of soil ordered as x and the total cost as y.
We are given two data points: (4, 220) and (7, 340).
To find the slope (m), we can use the formula:
m = (change in y) / (change in x)
m = (y2 - y1) / (x2 - x1)
Using the points (4, 220) and (7, 340):
m = (340 - 220) / (7 - 4)
m = 120 / 3
m = 40
Now we need to find the y-intercept (b). We can choose either of the given points and substitute the values of x, y, and m:
Using the point (4, 220):
y = mx + b
220 = 40 * 4 + b
220 = 160 + b
b = 220 - 160
b = 60
Therefore, the equation of the linear function that models the relationship between the number of cubic yards of soil ordered and the total cost is:
y = 40x + 60
To find the equation of the linear function that models the relationship between the number of cubic yards of soil ordered and the total cost, we can use the formula for the equation of a straight line:
y = mx + b
where:
- y represents the total cost
- x represents the number of cubic yards of soil ordered
- m represents the slope of the line
- b represents the y-intercept
To determine the slope, m, we can find the difference in total cost (y) for the two given data points:
Change in y = Total cost for this year - Total cost from last year
Change in x = Cubic yards ordered for this year - Cubic yards ordered from last year
Change in y = $340 - $220 = $120
Change in x = 7 - 4 = 3
Now, we can calculate the slope, m, by dividing the change in y by the change in x:
m = (Change in y) / (Change in x) = $120 / 3 = $40
Next, we need to determine the y-intercept, b. We can do this by substituting one of the given data points into the equation and solving for b. Let's use the first data point:
y = mx + b
$220 = $40 * 4 + b
$220 = $160 + b
To isolate b, we subtract $160 from both sides:
$220 - $160 = b
b = $60
Therefore, the equation of the linear function that models the relationship between the number of cubic yards of soil ordered (x) and the total cost (y) is:
y = $40x + $60