Intermediate Algebra, Chapter 3, 3.3, Section 3.3, Problem 36

Determine an equation of the line that satisfies the condition "through $(12,10)$; slope $1$".

(a) Write the equation in standard form.

Use the Point Slope Form of the equation of a line with $(x_1,y_1) = (12,10)$ and $m = 1$


$\begin{equation} \begin{aligned} y - y_1 =& m (x - x_1) && \text{Point Slope Form} \\ y - 10 =& 1 (x - 12) && \text{Substitute $x = 12, y = 10$ and $m=1$} \\ y - 10 =& x - 12 && \text{Distributive Property} \\ -x + y =& -12 + 10 && \text{Subtract each side by $(x-10)$} \\ -x + y =& -2 && \text{Standard Form} \\ \text{or} & && \\ x - y =& 2 && \end{aligned} \end{equation}$



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


$\begin{equation} \begin{aligned} -x + y =& -2 && \text{Standard Form} \\ y =& x - 2 && \text{Slope Intercept Form} \end{aligned} \end{equation}$