1.12.4 Writing Equations in Different Forms

Writing Equivalent Linear Forms

Write an equation of a line in all three forms that has a slope of 2 and a $y$-intercept of 7.

Slope-Intercept Form

• Slope-intercept form is $y=mx+b$ where m is the slope and $b$ is the $y$-intercept.

Substitute $m=2$ and $b=7$ to have the equation $y=2x+7$.

Standard Form

Write the same equation in standard form.

• Standard form is $Ax+By=C$.

• Remember that $A$ can not be negative and there should not be any fractions or decimals.

For this form, rearrange slope-intercept form to lead with the $x$ term:

Step 1 - Bring the $x$ and $y$ terms on the left side.

$y-2x=2x-2x+7$

Step 2 - Simplify.

$-2x+y=7$

Step 3 - Divide all terms by -1 so that $A$ is not negative.

$\frac{-2x}{-1}+\frac{y}{-1}=7-1$

Step 4 - Simplify.

$2x-y=-7$

Point-Slope Form

• Point-Slope form is $y-y_1=m(x-x_1)$.

• Substitute the slope in for $m$ and the point in for $(x_1,y_1)$.

$m=2$, $x_1=0$, $y_1=7$

$y-7=2(x-0)$

Notice this can easily be written in slope-intercept form by solving for $y$:

$\begin{array}{ccc}\hfill y-7& \hfill =\hfill & 2(x-0)\hfill \\ \hfill y-7& \hfill =\hfill & 2x-0\hfill \\ \hfill y-7+7& \hfill =\hfill & 2x-0+7\hfill \\ \hfill y& \hfill =\hfill & 2x+7\hfill \end{array}$

Standard Form to Slope-Intercept Form

Standard form can also be rearranged into slope-intercept form.

Write $3x-5y=14$ in slope-intercept form.

Work to solve for $y$.

Step 1 - Bring the $x$ term to the right side with the constant.

$3x-3x-5y=14-3x$

Step 2 - Simplify.

$-5y=14-3x$

Step 3 - Divide by the coefficient of $y$.

$\frac{-5y}{-5}=\frac{14}{5}-\frac{3}{5}x$

Step 4 - Simplify and write in the correct order.

$y=-\frac{3}{5}x+\frac{14}{5}$