Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. $\newline$ A librarian is expanding some sections of the city library. He buys books at a special price from a dealer who charges one price for any hardback book and another price for any paperback book. For the children's section, Mr. Crosby purchased $28$ new hardcover books and $38$ new paperback books, which cost a total of $\$506$ . He also purchased $58$ new hardcover books and $38$ new paperback books for the adult fiction section, spending a total of $\$926$ . What is the special price for each type of book? $\newline$ The special price is $\$\_\_\_\_$ for hardcover books and $\$\_\_\_\_$ for paperback books.

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Algebra 1

Solve a system of equations using elimination: word problems

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

\newline

A librarian is expanding some sections of the city library. He buys books at a special price from a dealer who charges one price for any hardback book and another price for any paperback book. For the children's section, Mr. Crosby purchased

28

new hardcover books and

38

new paperback books, which cost a total of

\$506

. He also purchased

58

new hardcover books and

38

new paperback books for the adult fiction section, spending a total of

\$926

. What is the special price for each type of book?

\newline

The special price is

\$\_\_\_\_

for hardcover books and

\$\_\_\_\_

for paperback books.

Define variables: Step $1$ : Define the variables for the prices of the books. $\newline$ Let $x$ be the price of one hardcover book and $y$ be the price of one paperback book.
Write equations: Step $2$ : Write the equations based on the information given. $\newline$ For the children's section: $28$ hardcover books and $38$ paperback books cost $\$506$ . $\newline$ Equation: $28x + 38y = 506$ $\newline$ For the adult fiction section: $58$ hardcover books and $38$ paperback books cost $\$926$ . $\newline$ Equation: $58x + 38y = 926$
Use elimination: Step $3$ : Use elimination to solve the system of equations. $\newline$ We can eliminate $y$ by subtracting the first equation from the second. $\newline$ $(58x + 38y) - (28x + 38y) = 926 - 506$ $\newline$ $30x = 420$
Solve for x: Step $4$ : Solve for x. $\newline$ Divide both sides by $30$ to find x. $\newline$ $x = \frac{420}{30}$ $\newline$ $x = 14$
Substitute and solve for y: Step $5$ : Substitute $x = 14$ back into one of the original equations to solve for y. $\newline$ Using the first equation: $28(14) + 38y = 506$ $\newline$ $392 + 38y = 506$ $\newline$ $38y = 506 - 392$ $\newline$ $38y = 114$ $\newline$ $y = \frac{114}{38}$ $\newline$ $y = 3$