Let's break the problem down and create a system of equations to describe the situation.

Defining variables:

• Let $x$ represent the number of kilograms of soap Leah needs to sell.
• Let $C(x)$ represent the total cost for Leah to make $x$ kilograms of soap.
• Let $S(x)$ represent the total sales Leah makes from selling $x$ kilograms of soap.

Step 1: Writing the cost equation

The total cost consists of two parts:

1. The initial cost of the equipment, which is $52.
2. The cost of the materials, which is $18 per kilogram.

Therefore, the total cost equation is:

$C(x) = 52 + 18x$

Step 2: Writing the sales equation

The sales income is $19 per kilogram. Therefore, the total sales equation is:

$S(x) = 19x$

Step 3: Setting up the condition

To determine when Leah covers the cost of the equipment, we set the sales equal to the cost:

$S(x) = C(x)$

Substitute the equations for $S(x)$ and $C(x)$:

$19x = 52 + 18x$

Step 4: Solving the equation

Now, solve for $x$:

$19x - 18x = 52$

$x = 52$

Step 5: Calculating total sales and costs

Once Leah sells 52 kilograms of soap:

• Total sales: $S(52) = 19 \times 52 = 988$
• Total costs: $C(52) = 52 + 18 \times 52 = 52 + 936 = 988$

Final Answer

• Once Leah sells 52 kilograms of soap, her sales and costs will both be $988.

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Would you like further explanation or a different method for solving the system? Here are a few related questions to explore:

1. How would the total costs and sales change if Leah sold the soap at a different price per kilogram?
2. How would the solution change if the initial equipment cost was higher or lower?
3. Can you determine the break-even point if the cost of materials increased?
4. How would this model change if Leah had other expenses besides the initial equipment cost and materials?
5. How does this model represent a typical "break-even" situation in business?

Tip: The break-even point is where your total sales exactly match your total costs, so you don’t have any profit or loss at that point.