The equation of the linear function that models the situation is: y = 0.75x + 7.50
Determining the Equation of the Linear Function
We can determine the equation of the linear function by finding the slope (m) and y-intercept (b) and using the formula:
y = mx + b
Step 1: Calculate the Slope (m)
The slope represents the change in cost per skein of embroidery floss. We can calculate it using the following formula:
m = (y2 - y1) / (x2 - x1)
Where:
y2 = Cost of project with 12 skeins = $16.50
y1 = Cost of project with 5 skeins = $11.25
x2 = Number of skeins with 12 skeins = 12
x1 = Number of skeins with 5 skeins = 5
Therefore:
m = ($16.50 - $11.25) / (12 - 5)
m = $5.25 / 7
m = 0.75
Step 2: Calculate the Y-Intercept (b)
The y-intercept represents the cost of the project with no skeins of embroidery floss. We can calculate it using the following formula:
b = y - mx
We can use the cost of the project with 5 skeins for this calculation.
b = $11.25 - (0.75)(5)
b = $11.25 - $3.75
b = $7.50
Step 3: Write the Equation
Now that we know the slope (m) and y-intercept (b), we can write the equation of the linear function:
y = mx + b
y = 0.75x + 7.50
Therefore, the equation of the linear function that models the cost of the cross-stitch project is y = 0.75x + 7.50, where:
y = Cost of the project
x = Number of skeins of embroidery floss