Intermediate Algebra, Chapter 3, 3.3, Section 3.3, Problem 56

Determine an equation of the line passing through the points $(5,-2)$ and $(-3,14)$.

(a) Write the equation in standard form.

Using the Slope Formula,

$\displaystyle m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{14-(-2)}{-3-5} = \frac{16}{-8} = -2$

Using Point Slope Form, where $m = -2$ and $(x_1,y_1) = (5,-2)$



$
\begin{equation}
\begin{aligned}

y - y_1 =& m(x - x_1)
&& \text{Point Slope Form}
\\
y - (-2) =& -2(x-5)
&& \text{Substitute } x = 5, y = -2 \text{ and } m = -2
\\
y + 2 =& -2x + 10
&& \text{Distributive Property}
\\
2x + y =& 10 - 2
&& \text{Add each side by $(2x - 2)$}
\\
2x + y =& 8
&& \text{Standard Form}

\end{aligned}
\end{equation}
$



(b) Write the equation in slope-intercept form.


$
\begin{equation}
\begin{aligned}

2x + y =& 8
&& \text{Standard Form}
\\
y =& -2x + 8
&& \text{Slope Intercept Form}

\end{aligned}
\end{equation}
$