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Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.

Ann wrote a business plan for an entrepreneurship class, and now she has to make bound copies. Ann could use a printer who charges a setup fee of $39 and $7 for every copy printed. Another possibility is to go to the office supply store, where she could pay an up-front fee of $36 and $10 per copy. There is a certain number of copies that makes the two options equivalent in terms of cost. How much would the copies cost? How many copies is that?

The cost is $
for
copies.

1 answer

Let \( x \) represent the number of copies Ann wants to make. We will create a system of equations based on the given costs from both options.

  1. The cost of printing at the printer: \[ C_1(x) = 39 + 7x \] Here, \( 39 \) is the setup fee and \( 7x \) is the cost for \( x \) copies.

  2. The cost of printing at the office supply store: \[ C_2(x) = 36 + 10x \] Here, \( 36 \) is the up-front fee, and \( 10x \) is the cost for \( x \) copies.

Next, we want to find the number of copies (\( x \)) that makes the two options equivalent in terms of cost. This occurs when \( C_1(x) = C_2(x) \):

\[ 39 + 7x = 36 + 10x \]

Now, we will solve for \( x \):

  1. Subtract \( 7x \) from both sides: \[ 39 = 36 + 3x \]

  2. Subtract \( 36 \) from both sides: \[ 3 = 3x \]

  3. Divide both sides by \( 3 \): \[ x = 1 \]

Now, substituting \( x = 1 \) back into either cost equation to find the total cost:

Using the printer's cost: \[ C_1(1) = 39 + 7(1) = 39 + 7 = 46 \]

Using the office supply store's cost: \[ C_2(1) = 36 + 10(1) = 36 + 10 = 46 \]

Both options cost the same at \( 1 \) copy.

Thus, the total cost is $46 for 1 copy.

The final answer is:

The cost is $ 46 for 1 copy.