If 6x = y+z and 4x = y-z, express z in terms of x

Question

\begin{align} 6x &= y+z\\ 4x &= y-z \end{align} How to express $z$ in terms of $x$?

I'm not 100% sure on how to solve in terms of x

Answer 1

Subtract the second equation from the first. So you have $6x -4x = (y+z) - (y-z).$

Answer 2

$$6x=y+z~........(1)$$ $$4x=y-z~........(2)$$ Substracting (2) form (1) we have, $$2x=2z$$ $$\implies z=x$$

Answer 3

While the answers given by absolute friend and Ebearr are elegant, their method of substracting the two equations from one another works only because the variable $y$ conveniently drops out upon doing so. This would not have been the case if your first equation would have been, say, $6x = 2y + z$.

Of course, in that case, one could have multiplied the second equation by $2$ before substracting the equations, but as you might imagine, with the equations becoming more complex, it might get hard to see what steps to take to make a variable cancel after substraction. This is why I'd like to show you another approach, which requires less thinking (but may result in more complicated calculations in some cases):

Take any of the two original equations. Let's choose $6x = y + z$. Solve that equation for $y$, acting as if $x$ and $z$ were known. Then you get $y = 6x - z$. Now you know what $y$ can be written as in terms of $x$ and $z$ alone, so that you can take the remaining equation, $4x = y - z$, and insert the result just obtained everywhere $y$ shows up: \begin{align} 4x &= y - z \\ &= (6x - z) - z \\&= 6x - 2z \end{align} Moving the $6x$ to the left-hand side of the equation, one has $-2x = -2z$, or in other words, $x=z$.