Write a system of equations to describe the situation below, solve using any method, and fill in the blanks. $\newline$ Lamar wrote a business plan for an entrepreneurship class, and now he has to make bound copies. Lamar could use a printer who charges a setup fee of $\$51$ and $\$7$ for every copy printed. Another possibility is to go to the office supply store, where he could pay an up-front fee of $\$24$ and $\$8$ 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? $\newline$ The cost is $\$$ _ for _ copies.

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Solve a system of equations using any method: word problems

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

\newline

Lamar wrote a business plan for an entrepreneurship class, and now he has to make bound copies. Lamar could use a printer who charges a setup fee of

\$51

and

\$7

for every copy printed. Another possibility is to go to the office supply store, where he could pay an up-front fee of

\$24

and

\$8

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?

\newline

The cost is

\$

_____ for _____ copies.

Set up equations: Set up the equations to represent the cost for each option. $\newline$ Printer's cost: Setup fee ( $\$51$ ) + $\$7$ per copy $\newline$ Office supply store's cost: Setup fee ( $\$24$ ) + $\$8$ per copy $\newline$ Let $x$ be the number of copies, and $C$ be the total cost for both options. $\newline$ Equation for the printer's cost: $C = 51 + 7x$ $\newline$ Equation for the office supply store's cost: $C = 24 + 8x$
Set costs equal: Since the costs are equivalent, we can set the two equations equal to each other to find the number of copies. $51 + 7x = 24 + 8x$
Solve for x: Solve for x by isolating the variable on one side. $\newline$ Subtract $7x$ from both sides: $\newline$ $51 + 7x - 7x = 24 + 8x - 7x$ $\newline$ $51 = 24 + x$ $\newline$ Subtract $24$ from both sides: $\newline$ $51 - 24 = x$ $\newline$ $x = 27$
Substitute $x$ into equation: Now that we have the number of copies ( $x$ ), we can find the total cost ( $C$ ) by substituting $x$ back into either of the original equations. $\newline$ Using the printer's cost equation: $C = 51 + 7x$ $\newline$ $C = 51 + 7(27)$
Calculate total cost: Calculate the total cost. $\newline$ $C = 51 + 7(27)$ $\newline$ $C = 51 + 189$ $\newline$ $C = 240$