Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Dr. Bradshaw just got her PhD, and she wants to print copies of her thesis in hardcover book format. She could use Belmont Printing, paying a setup fee of $\$51$ and $\$4$ for every book printed. Alternately, she could go through Norwood University, paying an up-front fee of $\$29$ and $\$6$ per book. It turns out that, given the number of books Dr. Bradshaw 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. Bradshaw just got her PhD, and she wants to print copies of her thesis in hardcover book format. She could use Belmont Printing, paying a setup fee of

\$51

and

\$4

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

\$29

and

\$6

per book. It turns out that, given the number of books Dr. Bradshaw 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: Let's define the variables. $\newline$ Let $x$ be the number of books Dr. Bradshaw wants to print. $\newline$ Let $y$ be the total cost for printing the books. $\newline$ Belmont Printing's cost can be represented as $y = 4x + 51$ . $\newline$ Norwood University's cost can be represented as $y = 6x + 29$ .
Set Up Equations: Set up the system of equations based on the given information. $\newline$ Equation for Belmont Printing: $y = 4x + 51$ $\newline$ Equation for Norwood University: $y = 6x + 29$
Use Substitution: Use substitution to solve for $x$ . Since both options cost the same amount, we can set the equations equal to each other: $4x + 51 = 6x + 29$
Solve for x: Solve for x. $\newline$ Subtract $4x$ from both sides: $51 = 2x + 29$ $\newline$ Subtract $29$ from both sides: $22 = 2x$ $\newline$ Divide both sides by $2$ : $x = 11$
Substitute for y: Substitute the value of $x$ back into one of the original equations to find $y$ . Using Belmont Printing's equation: $y = 4x + 51$ $y = 4(11) + 51$ $y = 44 + 51$ $y = 95$
Check Solution: Check the solution by substituting $x$ into Norwood University's equation. $y = 6x + 29$ $y = 6(11) + 29$ $y = 66 + 29$ $y = 95$ Since both equations give us the same $y$ value, our solution is correct.