Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. Band students at Silvergrove High School sell candy every year as a fundraiser. Last year, they sold 86 boxes of truffles and 87 boxes of peanut brittle, raising a total of $519. This year, they sold 62 boxes of truffles and 74 boxes of peanut brittle, from which they raised $408. 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 Silvergrove High School sell candy every year as a fundraiser. Last year, they sold 86 boxes of truffles and 87 boxes of peanut brittle, raising a total of $519. This year, they sold 62 boxes of truffles and 74 boxes of peanut brittle, from which they raised $408. 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 variables: Let's define two variables: let x be the amount the band earns from each box of truffles, and y be the amount the band earns from each box of peanut brittle. We can then write two equations based on the information given:  1. For last year's sales: 86x + 87y = 519 2. For this year's sales: 62x + 74y = 408  These two equations form our system of equations.Eliminate variable y: To solve this system using elimination, we need to eliminate one of the variables. We can do this by multiplying the equations by numbers that will make the coefficients of one of the variables the same. Let's try to eliminate y by finding a common multiple of 87 and 74. The least common multiple of 87 and 74 is 87 * 74, but we can use smaller multipliers to keep the numbers manageable. We can multiply the first equation by 74 and the second equation by 87 to get the coefficients of y to match:  74(86x + 87y) = 74 * 519 87(62x + 74y) = 87 * 408  This gives us: 74 * 86x + 74 * 87y = 74 * 519 87 * 62x + 87 * 74y = 87 * 408Perform multiplication: Now let's perform the multiplication:  6364x + 6438y = 38406 5394x + 6438y = 35496  We now have two new equations: 1. 6364x + 6438y = 38406 2. 5394x + 6438y = 35496Subtract equations: Next, we subtract the second equation from the first to eliminate y:  (6364x + 6438y) - (5394x + 6438y) = 38406 - 35496  This simplifies to: 970x = 2910Solve for x: Now we divide both sides by 970 to solve for x:  970x / 970 = 2910 / 970  x = 3Substitute x into equation: Now that we have the value for x, we can substitute it back into one of the original equations to solve for y. Let's use the first equation:  86x + 87y = 519  Substitute x = 3:  86(3) + 87y = 519  258 + 87y = 519Solve for y: Subtract 258 from both sides to solve for y:  87y = 519 - 258  87y = 261Final solution: Now we divide both sides by 87 to solve for y:  87y / 87 = 261 / 87  y = 3We have found that x = 3 and y = 3, which means the band earns $3 from each box of truffles and $3 from each box of peanut brittle.

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