Question 1150743
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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.<br>
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.<br>
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.<br><pre>
   a - 7b + 8c = 4  [1]
  8a + 4b -  c = 7  [2]

eliminate c....

   a - 7b + 8c =  4
 64a +32b - 8c = 56
 ------------------
 65a +25b      = 60

 13a + 5b = 12

  a = (12-5b)/13  [3]

eliminate a....

  8a -56b +64c = 32
  8a + 4b -  c =  7
  -----------------
     -60b +65c = 25

  -12b+13c = 5

  c = (12b+5)/13  [4]

Use [3] and [4] to evaluate a^2-b^2+c^2.</pre>

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.<br>