Question

Find the equation of the line
make sure its in standard form

Answer 1

keeping in mind that parallel lines have exactly the same slope, let's check for the slope of the equation above

`y=\stackrel{\stackrel{m}{\downarrow }}{\cfrac{4}{3}}x+10\qquad \impliedby \qquad \begin{array}ll \cline{1-1} slope-intercept~form\\ \cline{1-1} \\ y=\underset{y-intercept}{\stackrel{slope\qquad }{\stackrel{\downarrow }{m}x+\underset{\uparrow }{b}}} \\\\ \cline{1-1} \end{array}`

so we're really looking for the equation of a lline whose slope is 4/3 and it passes through (4 , -8) in standard form

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}{4}~,~\stackrel{y_1}{-8})\hspace{10em} \stackrel{slope}{m} ~=~ \cfrac{4}{3} \\\\\\ \begin{array}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}{(-8)}=\stackrel{m}{ \cfrac{4}{3}}(x-\stackrel{x_1}{4}) \implies y +8 = \cfrac{4}{3} ( x -4)`

`y+8=\cfrac{4}{3}x-\cfrac{16}{3}\implies \stackrel{\textit{multiplying both sides by }\stackrel{LCD}{3}}{3(y+8)=3\left( \cfrac{4}{3}x-\cfrac{16}{3} \right)}\implies 3y+24=4x-16 \\\\\\ 3y=4x-40\implies -4x+3y=-40\implies {\Large \begin{array}{llll} 4x-3y=40 \end{array}}`