How do you write an equation of a line given x-int of -4 and a y-int of 6?

Feb 9, 2017

$y = \textcolor{b l u e}{\frac{3}{2}} \left(x + \textcolor{red}{4}\right)$

Or

$\left(y - \textcolor{red}{6}\right) = \textcolor{b l u e}{\frac{3}{2}} x$

Or

$y = \textcolor{b l u e}{\frac{3}{2}} x + 6$

Explanation:

x-intercept of -4 gives us the point (-4, 0)

y-intercept of 6 gives us the point (0. 6)

We can use the point-slope formula to write equations for this line.
The slope can be found by using the formula: $m = \frac{\textcolor{red}{{y}_{2}} - \textcolor{b l u e}{{y}_{1}}}{\textcolor{red}{{x}_{2}} - \textcolor{b l u e}{{x}_{1}}}$

Where $m$ is the slope and ($\textcolor{b l u e}{{x}_{1} , {y}_{1}}$) and ($\textcolor{red}{{x}_{2} , {y}_{2}}$) are the two points on the line.

Substituting the values from the points and calculating the slope gives:

$m = \frac{\textcolor{red}{6} - \textcolor{b l u e}{0}}{\textcolor{red}{0} - \textcolor{b l u e}{- 4}}$

$m = \frac{\textcolor{red}{6} - \textcolor{b l u e}{0}}{\textcolor{red}{0} + \textcolor{b l u e}{4}} = \frac{6}{4} = \frac{2 \times 3}{2 \times 2} = \frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{2}}} \times 3}{\textcolor{red}{\cancel{\textcolor{b l a c k}{2}}} \times 2} = \frac{3}{2}$

The point-slope formula states: $\left(y - \textcolor{red}{{y}_{1}}\right) = \textcolor{b l u e}{m} \left(x - \textcolor{red}{{x}_{1}}\right)$

Where $\textcolor{b l u e}{m}$ is the slope and $\textcolor{red}{\left(\left({x}_{1} , {y}_{1}\right)\right)}$ is a point the line passes through.

Substituting the slope we calculated and the first point gives:

$\left(y - \textcolor{red}{0}\right) = \textcolor{b l u e}{\frac{3}{2}} \left(x - \textcolor{red}{- 4}\right)$

$y = \textcolor{b l u e}{\frac{3}{2}} \left(x + \textcolor{red}{4}\right)$

We can also substitute the slope we calculated and the second point giving:

$\left(y - \textcolor{red}{6}\right) = \textcolor{b l u e}{\frac{3}{2}} \left(x - \textcolor{red}{0}\right)$

$\left(y - \textcolor{red}{6}\right) = \textcolor{b l u e}{\frac{3}{2}} x$

We can also solve this equation for $y$ to give an equation in the slope-intercept form:

$y - \textcolor{red}{6} + 6 = \textcolor{b l u e}{\frac{3}{2}} x + 6$

$y - 0 = \textcolor{b l u e}{\frac{3}{2}} x + 6$

$y = \textcolor{b l u e}{\frac{3}{2}} x + 6$