# Solving Equations: Rewriting Formulas

## Rewriting a formula for a different variable

The area of a trapezoid is resented by the formula $A=\frac{1}{2}(b_1+b_2)h$.

I want draw several trapezoids, all with the same Area (to be decided later), as I change the bases I need to know what height to draw the trapezoid. To make the math easier I will re-write the formula for height ($h$).

\begin{align*} A &= \frac{1}{2}(b_1+b_2)h \\ \\ \color{blue}{2 *} A &= \color{blue}{2 *} \frac{1}{2}(b_1+b_2)h \\ \\ 2A &= (b_1+b_2)h \\ \\ \frac{2A}{\color{blue}{(b_1+b_2)}} &= \frac{(b_1+b_2)}{\color{blue}{(b_1+b_2)}}h \\ \\ \color{red}{\frac{2A}{(b_1+b_2)}} &= \color{red}{h} \\ \end{align*}

## Rewriting an equation for a different variable

Often when graphing an equation it is useful to solve for y
(also known as $y-intercept$ form.)

ex. Re-write the equation $3y-5x=12$ in $y-intercept$ form.

\begin{align*} 3y-5x &= 12 \\ \\ 3y-5x \color{blue}{-5x} &= \color{blue}{{}^-5x} + 12 \\ \\ 3y &= {}^-5x+12 \\ \\ \frac{3y}{\color{blue}{3}} &= \frac{{}^-5x}{\color{blue}{3}}+\frac{12}{\color{blue}{3}} \\ \\ \color{red}{y} &= \color{red}{\frac{{}^-5x}{3} + 4 } \end{align*}