System of Equations with Square Pattern Find 16x + 25y + 36z if: 
x + 4y + 9z = 10
4x + 9y + 16z = 120
9x + 16y + 25z = 1230
I tried using "brute force" and solving for each variable but the numbers are very large and messy ( I do not want to use a calculator). I tried factoring and creating a polynomial along the lines of 
a²x + (a+1)²y + (a+2)²z and using the given constants as roots, but that didn't work either. 
 A: One approach: find $a$, $b$, $c$ such that
$$
16x + 25y + 36z \equiv a(x + 4y + 9z) + b(4x + 9y + 16z) + c(9x + 16y + 25z)
$$
You do this by equating coefficients of $x,y,z$. You'll get three linear equations that you solve to get $a,b,c$. The solution turns out to be $a=1, b=-3, c=3$, so the intervening arithmetic is probably fairly simple, too.
Then, of course,
$$
16x + 25y + 36z = 10a + 120b + 1230c = 3340
$$
A: To get the solution you need to sum the equations different times, let's say
16 x + 25 y + 36 z = a * 1st equation + b * 2nd + c * 3rd  
From this, as variables are independent:
a + 4 b + 9 c = 16 (to get correct number of x in the main equation)
4 a + 9 b + 16 c = 25
9 a + 16 b + 25 c = 36  
Now let's multiply the first equation by 4 and 9 and subtract second and third ones from it respectively:
7 b + 20 c = 39
20 b + 56 c = 108  
Now it's easy to solve for b and c:
140 b + 400 c = 780
140 b + 392 c = 756  
8 c = 24
c = 3
b = -3
Replacing them to any of the first three equations:
a = 1 
So, using the solutions of the original equations:
16x + 25y + 36z = 10 a + 120 b + 1230 c = 3340
