Intermediate Algebra, Chapter 3, 3.3, Section 3.3, Problem 40

Determine an equation of the line that satisfies the condition "through $(7,-2)$; slope $\displaystyle \frac{1}{4}$".

(a) Write the equation in standard form.

Use the Point Slope Form of the equation of a line with $(x_1,y_1) = (7,-2)$ and $m = \displaystyle \frac{1}{4}$


$\begin{equation} \begin{aligned} y - y_1 =& m (x - x_1) && \text{Point Slope Form} \\ \\ y - (-2) =& \frac{1}{4} (x-7) && \text{Substitute $x = 7, y = -2$ and } m = \frac{1}{4} \\ \\ y + 2 =& \frac{1}{4}x - \frac{7}{4} && \text{Distributive Property} \\ \\ - \frac{1}{4}x + y =& - \frac{7}{4} - 2 && \text{Subtract each side by } \left( \frac{1}{4}x + 2 \right) \\ \\ - \frac{1}{4}x + y =& - \frac{15}{4} && \text{Standard Form} \\ \text{or} & &&& \\ x - 4y =& 15 && \end{aligned} \end{equation}$



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


$\begin{equation} \begin{aligned} - \frac{1}{4}x + y =& - \frac{15}{4} && \text{Standard Form} \\ \\ y =& \frac{1}{4}x - \frac{15}{4} && \text{Slope Intercept Form} \end{aligned} \end{equation}$