On one trip to the store, Heather bought 22 boxes of crackers and 88 packages of cheese for $23$23. On another trip to the store, she bought 88 boxes of crackers and 55 packages of cheese for $52$52.She determined the system of equations that represents the situation. 2x+8y=232𝑥+8𝑦=238x+5y=528𝑥+5𝑦=52 She graphed the equations, as shown. Assume the price for each item stays the same between visits. Between which 22 integers is the price of each item, in dollars?Enter your answer in each box.Price of box of crackers: Between $ and $Price of package of cheese: Between $ and $
Question
On one trip to the store, Heather bought 22 boxes of crackers and 88 packages of cheese for 23. On another trip to the store, she bought 88 boxes of crackers and 55 packages of cheese for 52.She determined the system of equations that represents the situation. 2x+8y=232𝑥+8𝑦=238x+5y=528𝑥+5𝑦=52 She graphed the equations, as shown. Assume the price for each item stays the same between visits. Between which 22 integers is the price of each item, in dollars?Enter your answer in each box.Price of box of crackers: Between Price of package of cheese: Between
Solution
The system of equations given is:
2x + 8y = 23 8x + 5y = 52
Where x represents the price of a box of crackers and y represents the price of a package of cheese.
To solve this system of equations, we can use substitution or elimination. Here, I'll use elimination.
First, multiply the first equation by 4 and the second equation by 2 to make the coefficients of x in both equations the same:
8x + 32y = 92 16x + 10y = 104
Subtract the second equation from the first:
-8x + 22y = -12
Solve for y:
y = -12/-22 = 0.5454...
So the price of a package of cheese is between 1.
Substitute y = 0.5454 into the first equation:
2x + 8(0.5454) = 23 2x + 4.3632 = 23 2x = 23 - 4.3632 2x = 18.6368 x = 18.6368/2 = 9.3184...
So the price of a box of crackers is between 10.
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