Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. 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? The band earns $_____ from each box of truffles and $_____ from each box of peanut brittle.

Question

Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.  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?  The band earns $_____ from each box of truffles and $_____ from each box of peanut brittle.

Bytelearn · Accepted Answer

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:  1. For last year's sales: $ 54t + 75p = 420 $ 2. For this year's sales: $ 53t + 83p = 431 $  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.  First equation multiplied by 53: $ (54t) \cdot 53 + (75p) \cdot 53 = 420 \cdot 53 $ Second equation multiplied by 54: $ (53t) \cdot 54 + (83p) \cdot 54 = 431 \cdot 54 $  Let's perform the multiplication.Perform Multiplication: After multiplying, we get:  First equation: $ 2862t + 3975p = 22260 $ Second equation: $ 2862t + 4482p = 23274 $  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:  $ (2862t + 4482p) - (2862t + 3975p) = 23274 - 22260 $  This simplifies to:  $ 4482p - 3975p = 23274 - 22260 $  Now we will calculate the difference.Calculate Difference: The difference gives us:  $ 507p = 1014 $  Now we will divide both sides by 507 to solve for $ p $.Solve for p: Dividing both sides by 507 gives us:  $ p = \frac{1014}{507} $  Calculating the division we get:  $ p = 2 $  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:  $ 54t + 75p = 420 $  Substituting $ p = 2 $ into the equation gives us:  $ 54t + 75 \cdot 2 = 420 $  Now we will calculate the value of $ t $.Calculate t: The equation becomes:  $ 54t + 150 = 420 $  Subtracting 150 from both sides gives us:  $ 54t = 270 $  Now we will divide both sides by 54 to solve for $ t $.Final Result: Dividing both sides by 54 gives us:  $ t = \frac{270}{54} $  Calculating the division we get:  $ t = 5 $  So, the band earns $5 from each box of truffles.

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