Question

Ariana has a home-based business making and selling scented soaps. She initially spent $100

to purchase soap-making equipment, and the materials for each pound of soap cost $8.
Ariana sells the soap for 10 per pound. Eventually, she will sell enough soap to cover the
cost of the equipment. How much soap will that be?
Write a system of equations, graph them, and type the solution.

Answer 1

To solve this problem, we can set up a system of equations based on the given information.

Let x represent the number of pounds of soap Ariana needs to sell to cover the cost of the equipment.

Equation 1: The cost of the equipment ($100) should be covered by selling enough soap.

Equation 2: The cost of materials for each pound of soap ($8) multiplied by the number of pounds of soap (x) should also be covered by selling enough soap.

Equation 3: The revenue generated from selling the soap (selling price per pound * number of pounds of soap) should be equal to the total cost.

Here are the equations:

1. 100 = 10x

2. 8x = 10x

3. 10x = 100

Now let's graph these equations to find the solution.

Graphing Equation 1: y = 10x

This equation represents the revenue generated from selling the soap.

Graphing Equation 2: y = 8x

This equation represents the cost of materials for each pound of soap.

Graphing Equation 3: y = 100

This equation represents the total cost of the equipment.

To find the point where all three lines intersect, we can plot the equations on a graph. The x-coordinate of the intersection point will give us the number of pounds of soap Ariana needs to sell.

After graphing the equations, we can see that the lines intersect at the point (10, 100). This means that Ariana needs to sell 10 pounds of soap to cover the cost of the equipment.

Therefore, Ariana needs to sell 10 pounds of soap to cover the cost of the equipment.