How do you solve for z in xz+y=1+z?

Mar 17, 2018

See a solution process below:

Explanation:

First, subtract $\textcolor{red}{y}$ and $\textcolor{b l u e}{z}$ from each side of the equation to isolate the $z$ terms while keeping the equation balanced:

$x z - \textcolor{b l u e}{z} + y - \textcolor{red}{y} = 1 - \textcolor{red}{y} + z - \textcolor{b l u e}{z}$

$x z - z + 0 = 1 - y + 0$

$x z - z = 1 - y$

Next, factor a $z$ out of each term on the left giving:

$z \left(x - 1\right) = 1 - y$

Now, divide each side of the equation by $\textcolor{red}{x - 1}$ to solve for $z$ while keeping the equation balanced:

$\frac{z \left(x - 1\right)}{\textcolor{red}{x - 1}} = \frac{1 - y}{\textcolor{red}{x - 1}}$

$\frac{z \textcolor{red}{\cancel{\textcolor{b l a c k}{\left(x - 1\right)}}}}{\cancel{\textcolor{red}{x - 1}}} = \frac{1 - y}{x - 1}$

$z = \frac{1 - y}{x - 1}$