System of Equations with Square Pattern

Question

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.

Answer 1

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

Answer 2

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, $$ 16x + 25y + 36z = 10a + 120b + 1230c = 3340 $$