# How do you solve T = (3/10)(Z - 12,000) for Z?

Jun 4, 2016

$Z = \frac{10}{3} \left(T + 3600\right)$

#### Explanation:

Given,

$T = \frac{3}{10} \left(Z - 12000\right)$

Use the distributive property, $\textcolor{red}{a} \left(\textcolor{b l u e}{b} + \textcolor{p u r p \le}{c}\right) = \textcolor{red}{a} \textcolor{b l u e}{b} + \textcolor{red}{a} \textcolor{p u r p \le}{c}$, to expand the right side.

$T = \frac{3}{10} \left(Z\right) + \frac{3}{10} \left(- 12000\right)$

$T = \frac{3}{10} Z + \frac{3 \times - 12000}{10}$

$T = \frac{3}{10} Z - \frac{36000}{10}$

$T = \frac{3}{10} Z - 3600$

Isolate for $Z$. Start by adding $3600$ to both sides.

$T \textcolor{w h i t e}{i} \textcolor{red}{+ 3600} = \frac{3}{10} Z - 3600 \textcolor{w h i t e}{i} \textcolor{red}{+ 3600}$

$T + 3600 = \frac{3}{10} Z$

Divide both sides by $\frac{3}{10}$.

$\textcolor{red}{\frac{\textcolor{b l a c k}{T + 3600}}{\frac{3}{10}}} = \textcolor{red}{\frac{\textcolor{b l a c k}{\frac{3}{10} Z}}{\frac{3}{10}}}$

$\textcolor{red}{\frac{\textcolor{b l a c k}{T + 3600}}{\frac{3}{10}}} = \textcolor{red}{\frac{\textcolor{b l a c k}{\textcolor{b l u e}{\cancel{\textcolor{b l a c k}{\frac{3}{10}}}} Z}}{\textcolor{b l u e}{\cancel{\textcolor{red}{\frac{3}{10}}}}}}$

Note: Recall that dividing by a fraction is the same as multiplying by its reciprocal!

$Z = \textcolor{g r e e n}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{\frac{10}{3} \left(T + 3600\right)} \textcolor{w h i t e}{\frac{a}{a}} |}}}$