Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. $\newline$ Band students at Greenpoint High School sell candy every year as a fundraiser. Last year, they sold $54$ boxes of truffles and $75$ boxes of peanut brittle, raising a total of $\$420$ . This year, they sold $53$ boxes of truffles and $83$ boxes of peanut brittle, from which they raised $\$431$ . How much does the band earn from each item? $\newline$ The band earns $\$\_$ from each box of truffles and $\$\_$ from each box of peanut brittle.

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Grade 8

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

Band students at Greenpoint High School sell candy every year as a fundraiser. Last year, they sold

54

boxes of truffles and

75

boxes of peanut brittle, raising a total of

\$420

. This year, they sold

53

boxes of truffles and

83

boxes of peanut brittle, from which they raised

\$431

. How much does the band earn from each item?

\newline

The band earns

\$\_

from each box of truffles and

\$\_

from each box of peanut brittle.

Define Equations: Let's denote the price of each box of truffles as $t$ and the price of each box of peanut brittle as $p$ . We can write two equations based on the information given: $\newline$ $1$ . For last year's sales: $54t + 75p = 420$ $\newline$ $2$ . For this year's sales: $53t + 83p = 431$ $\newline$ These two equations form our system of equations.
Elimination Method: To solve this system using elimination, we need to eliminate one of the variables. We can do this by multiplying the first equation by $53$ and the second equation by $54$ , so that when we subtract one equation from the other, the $t$ terms will cancel out. $\newline$ First equation multiplied by $53$ : $(54t) \cdot 53 + (75p) \cdot 53 = 420 \cdot 53$ $\newline$ Second equation multiplied by $54$ : $(53t) \cdot 54 + (83p) \cdot 54 = 431 \cdot 54$ $\newline$ Let's perform the multiplication.
Perform Multiplication: After multiplying, we get: $\newline$ First equation: $2862t + 3975p = 22260$ $\newline$ Second equation: $2862t + 4482p = 23274$ $\newline$ Now we will subtract the first equation from the second equation to eliminate $t$ .
Subtract Equations: Subtracting the first equation from the second gives us: $\newline$ $(2862t + 4482p) - (2862t + 3975p) = 23274 - 22260$ $\newline$ This simplifies to: $\newline$ $4482p - 3975p = 23274 - 22260$ $\newline$ Now we will calculate the difference.
Calculate Difference: The difference gives us: $\newline$ $507p = 1014$ $\newline$ Now we will divide both sides by $507$ to solve for $p$ .
Solve for p: Dividing both sides by $507$ gives us: $\newline$ $p = \frac{1014}{507}$ $\newline$ Calculating the division we get: $\newline$ $p = 2$ $\newline$ So, the band earns $2$ from each box of peanut brittle.
Substitute p into Equation: Now that we know the value of $ p $, we can substitute it back into one of the original equations to solve for $ t $. Let's use the first equation:$\newline$$ 54t + 75p = 420 $$\newline$Substituting $ p = 2 $ into the equation gives us:$\newline$$ 54t + 75 \cdot 2 = 420 $$\newline$Now we will calculate the value of $ t $.
Calculate t: The equation becomes:$\newline$$ 54t + 150 = 420 $$\newline$Subtracting $150$ from both sides gives us:$\newline$$ 54t = 270 $$\newline$Now we will divide both sides by $54$ to solve for $ t $.
Final Result: Dividing both sides by $54$ gives us:$\newline$$ t = \frac{270}{54} $$\newline$Calculating the division we get:$\newline$$ t = 5 $$\newline$So, the band earns $5$ from each box of truffles.