How to solve this equation with three variables arithmetically?

Question

How can we find the sum of $x$, $y$, and $z$ arithmetically? $$x+y=6$$ $$z+y=10$$ $$x+y+z=?$$

Some things I have tried: $$y=6-x$$ $$y=10-z$$ $$6-x=10-z$$ $$z-x=4$$ or $$x+2y+z=16$$ or $$x=6-y$$ $$z=10-y$$ $$x+z=6-y+10-y$$ $$x+z=16-2y$$ Will these lead me anywhere?

Answer 1

Just having $x+y=6$ and $y+z=10$ is not enough information to determine $x+y+z$.

For example, we could have $(x,y,z)=(6,0,10)$ with sum $16$, or $(x,y,z)=(-10,16,-6)$ with sum $0$.

In general you can get any sum you want by setting $(x,y,z)=(t-10,16-t,t-6)$, giving a sum of $t$.

Even if, for example, you want $x$, $y$ and $z$ all to be positive, you can still get any $t$ strictly between $10$ and $16$.

Answer 2

Another way you can solve this is using matrix row operations. Place the systems of equations in to a 3x4 matrix and solve.

$$ \left[ \begin{array}{ccc|c} 1&2&0&6\\ 0&1&1&10\\ 1&1&1&a \end{array} \right] $$

Where $a = ?$ And the columns represent $x,y, z$ respectively.