standard form for a linear equation means
• all coefficients must be integers, no fractions
• only the constant on the right-hand-side
• all variables on the left-hand-side, sorted
• "x" must not have a negative coefficient
[tex](\stackrel{x_1}{6}~,~\stackrel{y_1}{7})\qquad (\stackrel{x_2}{-3}~,~\stackrel{y_2}{10}) \\\\\\ \stackrel{slope}{m}\implies \cfrac{\stackrel{rise} {\stackrel{y_2}{10}-\stackrel{y1}{7}}}{\underset{run} {\underset{x_2}{-3}-\underset{x_1}{6}}} \implies \cfrac{10 -7}{-3 -6} \implies \cfrac{ 3 }{ -9 }\implies -\cfrac{1}{3}[/tex]
[tex]\begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{7}=\stackrel{m}{-\cfrac{1}{3}}(x-\stackrel{x_1}{6})\implies y-7=-\cfrac{x}{3}+2 \\\\\\ \stackrel{\textit{multiplying both sides by }\stackrel{LCD}{3}}{3(y-7)=3\left( -\cfrac{x}{3}+2\right)}\implies 3y-21=-x+6 \\\\\\ 3y=-x+27\implies \blacksquare~~ x+3y=27 ~~\blacksquare[/tex]
