Question

Use the giving conditions to write an equation for the line in point slope form and un slope intercept form.

Passing through (9,-2) and perpendicular to the line whose equation is y=1/3x+5

Answer 1

keeping in mind that perpendicular lines have negative reciprocal slopes, let's check for the slope of the equation above

`y=\stackrel{\stackrel{m}{\downarrow }}{\cfrac{1}{3}}x+5\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} \\\\[-0.35em] ~\dotfill`

`\stackrel{~\hspace{5em}\textit{perpendicular lines have \underline{negative reciprocal} slopes}~\hspace{5em}} {\stackrel{slope}{ \cfrac{1}{3}} ~\hfill \stackrel{reciprocal}{\cfrac{3}{1}} ~\hfill \stackrel{negative~reciprocal}{-\cfrac{3}{1} \implies -3}}`

so we're really looking for the equation of a line whose slope is -3 and it passes through (9 , -2)

`(\stackrel{x_1}{9}~,~\stackrel{y_1}{-2})\hspace{10em} \stackrel{slope}{m} ~=~ -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}{(-2)}=\stackrel{m}{-3}(x-\stackrel{x_1}{9}) \implies y +2 = -3 ( x -9) \\\\\\ y+2=-3x+27\implies {\Large \begin{array}{llll} y=-3x+25 \end{array}}`

Answer 2

y + 2 = - 3(x - 9) and y = - 3x + 25

Explanation:

the equation of a line in slope- intercept form is

y = mx + c ( m is the slope and c the y- intercept )

given

y =
`(1)/(3)` x + 5 ← in slope- intercept form

with slope m =
`(1)/(3)`

given a line with slope m then the slope of a line perpendicular to it is

`m_(perpendicular)` = -
`(1)/(m)` = -
`(1)/((1)/(3) )` = - 3

the equation of a line in point- slope form is

y - b = m(x - a)

where m is the slope and (a, b ) a point on the line

here m = - 3 and (a, b ) = (9, - 2 ) , then

y - (- 2) = - 3(x - 9) , that is

y + 2 = - 3(x - 9) ← in point- slope form

distribute the parenthesis by - 3

y + 2 = - 3x + 27 ( subtract 2 from both sides )

y = - 3x + 25 ← in slope- intercept form