Question

Part 4:

7. What is the point slope equation of line that passes throug
point (2,-3) and has the same y intercept as y = 5x + 2?

Answer 1

to get the equation of any straight line, we simply need two points off of it, hmmm let's take a peek at the y-intercept of the line above

`y = 5x \stackrel{\stackrel{b}{\downarrow }}{+2} \qquad \impliedby \begin{array}c \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}\qquad \implies \stackrel{y-intercept}{(0~~,~~2)}`

so we're really looking for the equation of a line that passes through (2 , -3) and (0 , 2)

`(\stackrel{x_1}{2}~,~\stackrel{y_1}{-3})\qquad (\stackrel{x_2}{0}~,~\stackrel{y_2}{2}) ~\hfill \stackrel{slope}{m}\implies \cfrac{\stackrel{rise} {\stackrel{y_2}{2}-\stackrel{y1}{(-3)}}}{\underset{run} {\underset{x_2}{0}-\underset{x_1}{2}}} \implies \cfrac{2 +3}{-2} \implies -\cfrac{ 5 }{ 2 }`

`\begin{array}c \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{ 5 }{ 2 }}(x-\stackrel{x_1}{2}) \implies y +3= -\cfrac{ 5 }{ 2 } (x -2)`