Question

Write the equation of the line, in all three forms, that passes through point

(-10,-3) and is parallel to y +=(x-1).
Point-Slope Form
Slope-Intercept Form
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+\cfrac{1}{4}= ~~ \stackrel{\stackrel{m}{\downarrow }}{\cfrac{1}{4}} ~~ (x-1)\qquad \impliedby \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}`

so we're really looking for the equation of a line whose slope is 1/4 and that it passes through (-10 , -3)

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}{-10}~,~\stackrel{y_1}{-3})\hspace{10em} \stackrel{slope}{m} ~=~ \cfrac{1}{4} \\\\\\ \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}{(-3)}=\stackrel{m}{ \cfrac{1}{4}}(x-\stackrel{x_1}{(-10)}) \implies \stackrel{\textit{\LARGE point-slope}}{y +3= \cfrac{1}{4} (x +10)}`

`y+3=\cfrac{1}{4}x+\cfrac{5}{2}\implies y=\cfrac{1}{4}x+\cfrac{5}{2}-3\implies \stackrel{\textit{\LARGE slope-intercept}}{y=\cfrac{1}{4}x-\cfrac{1}{2}} \\\\\\ \stackrel{\textit{multiplying both sides by }\stackrel{LCD}{4}}{4(y)=4\left( \cfrac{1}{4}x-\cfrac{1}{2} \right)}\implies 4y=x-2 \\\\\\ -x+4y=-2\implies \stackrel{\textit{\LARGE standard}}{x-4y=2}`