# How to manually solve this system of four equations?

Sorry for coming up with a "do-it-for-me"-question, but I can't think of another way to get help with manually solving the following system of four equations:

1. $50=x*\cos(\beta)-y*\cos(\alpha)$

2. $0=x*\sin(\beta)-y*\sin(\alpha)$

3. $0=x*\sin(\gamma)+y*\cos(\alpha)$

4. $45=x*\cos(\gamma)-y*\sin(\alpha)$

$x$, $y$, $\beta$ and $\gamma$ are unknown.

$\alpha$ is a known constant with the given value of $63.5°$.

I tried all approaches I could think of and always got stuck on one equation with two unknown angels, for example $50\tan\beta*(\sin\alpha\tan\gamma+\cos\alpha) = 45\tan\gamma*(\cos\alpha\tan\beta-\sin\alpha)$.

EDIT: This is the numerical solution of WolframAlpha, but, as stated above, I need to do it manually.

http://www.wolframalpha.com/input/?i=50%3Dxcos%28beta%29-ycos%2863.5*pi%2F180%29%2C+0%3Dxsin%28beta%29-ysin%2863.5*pi%2F180%29%2C+0%3Dxsin%28gamma%29%2Bycos%2863.5*pi%2F180%29%2C+45%3Dxcos%28gamma%29-ysin%2863.5*pi%2F180%29%2C+solve+for+x%2Cy%2Cbeta%2Cgamma

## 1 Answer

In all the equations move the y term to the LHS. Square the first two equations and add. Use Sin^2 + Cos^2=1. So have equation for x^2 & terms in y & y^2. Do the same with 3rd & 4th equations. Eliminate x^2 from the resultant equations to give quadratic in y which you can solve. From here you have x and so on and so on.

• Squaring makes sense, thanks a lot! – MaxD Feb 4 '15 at 13:30