Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. $\newline$ The Ashland High School Science Department is purchasing new earth science and physics textbooks this year. Ms. Walton has requested $80$ earth science textbooks and $92$ physics textbooks for all of her classes, which costs the department a total of $\$7,744$ . Mr. Castro has asked for $80$ earth science textbooks and $76$ physics textbooks, which will cost a total of $\$6,912$ . How much do the textbooks cost? $\newline$ Earth science textbooks cost $\$$ apiece and physics textbooks cost $\$$ apiece.

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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

The Ashland High School Science Department is purchasing new earth science and physics textbooks this year. Ms. Walton has requested

80

earth science textbooks and

92

physics textbooks for all of her classes, which costs the department a total of

\$7,744

. Mr. Castro has asked for

80

earth science textbooks and

76

physics textbooks, which will cost a total of

\$6,912

. How much do the textbooks cost?

\newline

Earth science textbooks cost

\$

____ apiece and physics textbooks cost

\$

____ apiece.

Define Costs: Let's denote the cost of one earth science textbook as $x$ dollars and the cost of one physics textbook as $y$ dollars. We can write two equations based on the information given: $\newline$ For Ms. Walton's request: $80$ earth science textbooks and $92$ physics textbooks cost a total of $\$7,744$ . $\newline$ For Mr. Castro's request: $80$ earth science textbooks and $76$ physics textbooks cost a total of $\$6,912$ .
Write Equations: Translate the information into a system of equations: $\newline$ Ms. Walton's order: $80x + 92y = 7,744$ $\newline$ Mr. Castro's order: $80x + 76y = 6,912$
Eliminate Variable: To use elimination, we need to eliminate one of the variables. Since the coefficient of $x$ is the same in both equations, we can eliminate $x$ by subtracting the second equation from the first. $\newline$ $(80x + 92y) - (80x + 76y) = 7,744 - 6,912$
Subtract Equations: Perform the subtraction to find the value of $y$ : $80x + 92y - 80x - 76y = 7,744 - 6,912$ $92y - 76y = 7,744 - 6,912$ $16y = 832$ $y = \frac{832}{16}$ $y = 52$
Find y Value: Now that we have the value of $y$ , we can substitute it back into one of the original equations to find the value of $x$ . Let's use Ms. Walton's order: $\newline$ $80x + 92(52) = 7,744$
Substitute y: Calculate the value of $x$ : $80x + 4,784 = 7,744$ $80x = 7,744 - 4,784$ $80x = 2,960$ $x = \frac{2,960}{80}$ $x = 37$
Calculate $x$ : We have found the values of $x$ and $y$ , which represent the cost of earth science and physics textbooks, respectively: $\newline$ $x = 37$ (cost of one earth science textbook) $\newline$ $y = 52$ (cost of one physics textbook)