Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Russell wrote a business plan for an entrepreneurship class, and now he has to make bound copies. Russell could use a printer who charges a setup fee of $\$38$ and $\$5$ for every copy printed. Another possibility is to go to the office supply store, where he could pay an up-front fee of $\$36$ and $\$7$ per copy. There is a certain number of copies that makes the two options equivalent in terms of cost. How many copies is that? How much would the copies cost? $\newline$ For $\_\_\_\_\_$ copies, the cost is $\$\_\_\_\_\_$ .

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Algebra 1

Solve a system of equations using substitution: word problems

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

\newline

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

\$38

and

\$5

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

\$36

and

\$7

per copy. There is a certain number of copies that makes the two options equivalent in terms of cost. How many copies is that? How much would the copies cost?

\newline

For

\_\_\_\_\_

copies, the cost is

\$\_\_\_\_\_

Define Variables: Step $1$ : Let's define the variables. $\newline$ Let $x$ be the number of copies Russell needs to print. $\newline$ Let $y$ be the total cost for printing $x$ copies.
Cost Equation for Printer: Step $2$ : Write the equation for the cost of using the printer. $\newline$ The printer charges a setup fee of $\$38$ and $\$5$ for every copy. $\newline$ So the cost equation for the printer is: $y = 5x + 38$ .
Cost Equation for Office Supply Store: Step $3$ : Write the equation for the cost of using the office supply store. $\newline$ The office supply store charges an up-front fee of $\$36$ and $\$7$ per copy. $\newline$ So the cost equation for the office supply store is: $y = 7x + 36$ .
Set Equations Equal: Step $4$ : Set the two equations equal to each other to find the number of copies where the costs are equivalent. $\newline$ $5x + 38 = 7x + 36$
Solve for x: Step $5$ : Solve for x by isolating the variable. $\newline$ $5x - 7x = 36 - 38$ $\newline$ $-2x = -2$ $\newline$ $x = -2 / -2$ $\newline$ $x = 1$
Substitute $x$ for Total Cost: Step $6$ : Substitute the value of $x$ back into either of the original equations to find the total cost $y$ . Using the printer's cost equation: $y = 5x + 38$ $y = 5(1) + 38$ $y = 5 + 38$ $y = 43$
Verify Solution: Step $7$ : Verify the solution by substituting $x$ into the office supply store's cost equation. $\newline$ Using the office supply store's cost equation: $y = 7x + 36$ $\newline$ $y = 7(1) + 36$ $\newline$ $y = 7 + 36$ $\newline$ $y = 43$