Question 114129: A football team is planning a banquet. It offers 2 types of meals. The first one is a turkey plate for $8.00 the second is a roast beef plate for $12.00. The total budget is $980.00 for 100 people. X is the turkey and Y is the roast.
X + Y = 100
8
X +12Y =980
a) solve the system of equations by elimination.
b) How many turkey plates are sold?
c) How many roast beef plates are sold?
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! Given:
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X + Y = 100
8X +12Y =980
.
To solve these two equations by elimination, either the terms in the X column or the terms
in the Y column must be equal in size. Let's make the terms in the X column equal. We can
do that by multiplying the top equation (all terms on both sides) by 8. When we do that multiplication
the two equations are:
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8X + 8Y = 800
8X +12Y = 980
.
Now if we subtract vertically in the columns the 8X from the bottom equation is subtracted
from the 8X in the top equation ... leaving 0X which means the X variable is now gone.
The 12Y from the bottom equation is taken from the 8Y in the top equation leaving
-4Y. And on the other side of the equation the 980 from the bottom equation is subtracted
from the 800 in the top equation resulting in -180. So after vertically subtracting
the two equations we are left with the equation:
.
-4Y = -180
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Dividing both sides of this equation by -4 (which is the multiplier of the Y), we have:
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Y = -180/-4 = 180/4 = 45
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So 45 roast beef plates are sold. [This is the answer to part c.]
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But a total of 100 plates were sold. So the remaining 55 plates have to be the number
of turkey plates sold. [This is the answer to part b.]
.
We could also have solved this problem by eliminating the Y column of the two equations.
Starting with the two original equations:
.
X + Y = 100
8X +12Y =980
.
we could have multiplied all terms of the top equation by 12 to make the equations
become:
.
12X +12Y =1200
8X + 12Y = 980
.
When you subtract these two equations vertically, the Y terms cancel out and the resulting
equation is:
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4X = 220
.
Dividing both sides by 4 results in:
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X = 220/4 = 55
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Again we have found that X, the number of turkey plates, is 55, but this time we found the
answer to part b of the problem first ... in the order asked for by the problem.
.
The answer to part c, the number of roast beef plates sold, is then found by subtracting
the number of turkey plates (55) from 100 (total plates sold) to again get 45 as the answer.
.
Hope this helps you to understand the problem and how to solve it by elimination.
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