Find the value of $a^2-b^2+c^2$ Let $a$, $b$, and $c$ be real numbers such that $a − 7b + 8c = 4$ and $8a + 4b − c = 7$. What is the
value of $a^2-b^2+c^2$ ?

Given $a + 8c= 7b + 4$ and $8a - c = -4b + 7$
Now Squaring both equation, We get 
\begin{align}a^2 + 64c^2 + 16ac&=49b^2 + 16 + 56b\tag{1} \\
64a^2 + c^2 - 16ac&=16b^2 + 49 - 56b\tag{2}\end{align} 
Summing these two equations gives 
$$ 65a^2 + 65c^2 = 65b^2 + 65 \Rightarrow  a^2 - b^2 + c^2 = 1 $$
