System of linear equations with four unkowns

Question

I have no idea how to solve this system of equation : $$\begin{align}u+v+w&=7 \\v+w+x&=-8 \\w+x+u&=5 \\x+u+v&=-10\end{align}$$

I usually use the addition/substraction method, but it leads no where in this case...

A little hint would be helpful.

Answer 1

To use the addition/subtraction method as you call it, you often need to add/subtract more than just two of the equations together when there are more than two variables and equations involved. Here, because of the symmetry, we can do something much easier and more concrete. Add all four of the equations together to get $3u+3v+3w+3x = -6$, giving

$$ u + v + w + x = -2$$

Subtract from this equation each given equation, and in turn you will get the answer for each variable. Subtracting the first given equation, for example, from the above equation you get $x=-9$, and so on.