Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. $\newline$ The Richmond High School Science Department is purchasing new earth science and physics textbooks this year. Ms. Lee has requested $83$ earth science textbooks and $65$ physics textbooks for all of her classes, which costs the department a total of $\$7,072$ . Mr. Barrett has asked for $83$ earth science textbooks and $83$ physics textbooks, which will cost a total of $\$8,134$ . How much do the textbooks cost? $\newline$ Earth science textbooks cost $\$\_$ apiece and physics textbooks cost $\$\_$ apiece.

View step-by-step help

Home

Math Problems

Grade 8

Solve a system of equations using elimination: word problems

Full solution

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

\newline

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

83

earth science textbooks and

65

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

\$7,072

. Mr. Barrett has asked for

83

earth science textbooks and

83

physics textbooks, which will cost a total of

\$8,134

. How much do the textbooks cost?

\newline

Earth science textbooks cost

\$\_

apiece and physics textbooks cost

\$\_

apiece.

Define variables: Let's define two variables: let $x$ be the cost of one earth science textbook, and $y$ be the cost of one physics textbook.
Write equations: We can write two equations based on the information given. For Ms. Lee's request, the equation is $83x + 65y = 7,072$ . For Mr. Barrett's request, the equation is $83x + 83y = 8,134$ .
Use elimination: To use elimination, we need to make the coefficients of one of the variables the same in both equations. The coefficients of $x$ are already the same, so we can subtract the first equation from the second to eliminate $x$ .
Subtract equations: Subtracting the first equation from the second gives us $83x + 83y - (83x + 65y) = 8,134 - 7,072$ .
Simplify equation: This simplifies to $83y - 65y = 8,134 - 7,072$ , which further simplifies to $18y = 1,062$ .
Find cost of physics textbook: Dividing both sides of the equation by $18$ gives us $y = 1,062 / 18$ , which simplifies to $y = 59$ . This means that each physics textbook costs $\$59$ .
Substitute value of y: Now that we know the cost of each physics textbook, we can substitute $y = 59$ into one of the original equations to find $x$ . Let's use the first equation: $83x + 65(59) = 7,072$ .
Solve for x: Substituting the value of $y$ into the equation gives us $83x + 3,835 = 7,072$ .
Solve for x: Substituting the value of $y$ into the equation gives us $83x + 3,835 = 7,072$ . Subtracting $3,835$ from both sides of the equation gives us $83x = 7,072 - 3,835$ , which simplifies to $83x = 3,237$ .
Solve for x: Substituting the value of $y$ into the equation gives us $83x + 3,835 = 7,072$ . Subtracting $3,835$ from both sides of the equation gives us $83x = 7,072 - 3,835$ , which simplifies to $83x = 3,237$ . Dividing both sides of the equation by $83$ gives us $x = \frac{3,237}{83}$ , which simplifies to $x = 39$ . This means that each earth science textbook costs $\$39$ .