Question

Find the standard equation of a line that passes through these two points: (-3, 4) and (2, 2)

Answer 1

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

`(\stackrel{x_1}{-3}~,~\stackrel{y_1}{4})\qquad (\stackrel{x_2}{2}~,~\stackrel{y_2}{2})~\hfill \stackrel{slope}{m}\implies \cfrac{\stackrel{rise} {\stackrel{y_2}{2}-\stackrel{y1}{4}}}{\underset{run} {\underset{x_2}{2}-\underset{x_1}{(-3)}}} \implies \cfrac{2 -4}{2 +3} \implies \cfrac{ -2 }{ 5 }`

`\begin{array} \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}{4}=\stackrel{m}{-\cfrac{2}{5}}(x-\stackrel{x_1}{(-3)})\implies y-4=-\cfrac{2}{5}(x+3) \\\\\\ \stackrel{\textit{multiplying both sides by }\stackrel{LCD}{5}}{5(y-4)~~ = ~~5\left( -\cfrac{2}{5}(x+3) \right)}\implies 5y-20=-2(x+3) \\\\\\ 5y-20=-2x-6\implies 2x+5y-20=-6\implies 2x+5y=14`