Question

The three points (2,5),(3,8) and (-1,y) belong to the same line,find the value of y

Answer 1

well, we know the points are all colinear, so the slope from the 1st to the 2nd, will be the same as from the 2nd to the 3rd, same exact slope, hell let's check what's up

`(\stackrel{x_1}{2}~,~\stackrel{y_1}{5})\qquad (\stackrel{x_2}{3}~,~\stackrel{y_2}{8}) ~\hfill \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{8}-\stackrel{y1}{5}}}{\underset{\textit{\large run}} {\underset{x_2}{3}-\underset{x_1}{2}}} \implies \cfrac{ 3 }{ 1 } \implies 3`

ahaaa! that means that the slope of (3 , 8) and (-1 , y) is 3, hmmmm

`(\stackrel{x_1}{3}~,~\stackrel{y_1}{8})\qquad (\stackrel{x_2}{-1}~,~\stackrel{y_2}{y}) \\\\\\ \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{y}-\stackrel{y1}{8}}}{\underset{\textit{\large run}} {\underset{x_2}{-1}-\underset{x_1}{3}}} ~~ = ~~\stackrel{\stackrel{\textit{\small slope}}{\downarrow }}{ 3 }\implies \cfrac{y-8}{-4}=3 \\\\\\ y-8=-12\implies \boxed{y=-4}`