Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Dr. Sloan just got her PhD, and she wants to print copies of her thesis in hardcover book format. She could use Silvergrove Printing, paying a setup fee of $\$43$ and $\$4$ for every book printed. Alternately, she could go through Weston University, paying an up-front fee of $\$28$ and $\$9$ per book. It turns out that, given the number of books Dr. Sloan wants to print, the two options cost the same amount. What is the amount? How many books is that? $\newline$ The cost is $\$\_\_\_\_$ for $\_\_\_\_$ books.

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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

Dr. Sloan just got her PhD, and she wants to print copies of her thesis in hardcover book format. She could use Silvergrove Printing, paying a setup fee of

\$43

and

\$4

for every book printed. Alternately, she could go through Weston University, paying an up-front fee of

\$28

and

\$9

per book. It turns out that, given the number of books Dr. Sloan wants to print, the two options cost the same amount. What is the amount? How many books is that?

\newline

The cost is

\$\_\_\_\_

for

\_\_\_\_

books.

Define Variables and Equations: Let's define the variables and write the system of equations. Let $x$ be the number of books Dr. Sloan wants to print, and let $y$ be the total cost for printing the books. $\newline$ For Silvergrove Printing, the cost is $\$43$ plus $\$4$ per book, so the equation is: $\newline$ $y = 4x + 43$ $\newline$ For Weston University, the cost is $\$28$ plus $\$9$ per book, so the equation is: $\newline$ $y = 9x + 28$ $\newline$ Since the costs are the same, we can set the two equations equal to each other: $\newline$ $4x + 43 = 9x + 28$
Set Equations Equal: Now, we will solve for $x$ using substitution. We can subtract $4x$ from both sides to get the $x$ terms on one side: $\newline$ $4x + 43 - 4x = 9x + 28 - 4x$ $\newline$ This simplifies to: $\newline$ $43 = 5x + 28$
Solve for x: Next, we subtract $28$ from both sides to isolate the term with $x$ : $\newline$ $43 - 28 = 5x + 28 - 28$ $\newline$ This simplifies to: $\newline$ $15 = 5x$
Isolate x Term: Now, we divide both sides by $5$ to solve for $x$ : $\newline$ $15 \div 5 = 5x \div 5$ $\newline$ This gives us: $\newline$ $x = 3$
Calculate x: We have found that Dr. Sloan wants to print $3$ books. Now we need to find the total cost, $y$ . We can substitute $x$ back into either of the original equations. Let's use the Silvergrove Printing equation: $\newline$ $y = 4x + 43$ $\newline$ $y = 4(3) + 43$
Substitute $x$ into Equation: Now, we calculate the cost: $\newline$ $y = 4(3) + 43$ $\newline$ $y = 12 + 43$ $\newline$ $y = 55$
Calculate Total Cost: So, the cost is $\$55$ for $3$ books. This is the final answer.