Question 1150743: Let a, b, and c be real numbers such that a - 7b + 8c = 4 and 8a + 4b - c = 7. Then a^2 - b^2 + c^2 is what?
Answer by greenestamps(13200)   (Show Source):
You can put this solution on YOUR website!
We have three unknowns and only two equations, so we can't solve to find the values of a, b, and c. But the problem doesn't ask us to do that; it only asks us to find the value of a^2-b^2+c^2.
Use elimination between the two equations to eliminate a, giving you an equation relating b and c; then use elimination again to eliminate c, giving you an equation relating b and c.
Then use those expressions to evaluate a^2-b^2+c^2. It turns out all the variable terms cancel, leaving you with a numerical value for the expression.
a - 7b + 8c = 4 [1]
8a + 4b - c = 7 [2]
eliminate c....
a - 7b + 8c = 4
64a +32b - 8c = 56
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65a +25b = 60
13a + 5b = 12
a = (12-5b)/13 [3]
eliminate a....
8a -56b +64c = 32
8a + 4b - c = 7
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-60b +65c = 25
-12b+13c = 5
c = (12b+5)/13 [4]
Use [3] and [4] to evaluate a^2-b^2+c^2.
I'll leave that to you. All the variable terms cancel, leaving you with what you want -- a numerical value for a^2-b^2+c^2.
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