# How do you solve for a in ax+ z = aw-y?

Apr 12, 2018

See a solution process below:

#### Explanation:

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

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

$a x - a w + 0 = 0 - y - z$

$a x - a w = - y - z$

Next, factor an $a$ from each term on the left side of the equation:

$a \left(x - w\right) = - y - z$

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

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

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

$a = \frac{- y - z}{x - w}$

We can then multiply the right side of the equation by a form of $1$ to rewrite the expression as:

$a = \frac{- 1}{-} 1 \times \frac{- y - z}{x - w}$

$a = \frac{- 1 \left(- y - z\right)}{- 1 \left(x - w\right)}$

$a = \frac{y + z}{- x + w}$

$a = \frac{y + z}{w - x}$