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.