Solution assignment 10 Linear functions and graphs

The line $l$ goes through the points $P(1,2)$ and $Q(-1,-1)$ . For which value of $r$ is the line:

$m: y=rx-3$

parallel to $l$ ?

Parallel lines have equal slopes. The slope of $l$ is:

$\displaystyle\frac{y_Q-y_P}{x_Q-x_P}=\displaystyle\frac{-1-2}{-1-1}=\displaystyle\frac{-3}{-2}=\displaystyle\frac{3}{2}$

The equation of $l$ is:

$y-2=\displaystyle\frac{3}{2}(x-1)$
$y=\displaystyle\frac{3}{2}x+\displaystyle\frac{1}{2}$

The line $l$ is parallel to $m$ if:

$r=\displaystyle\frac{3}{2}$ and $\displaystyle\frac{1}{2}\neq-3$

Both conditions are satisfied and thus the line $l$ is parallel to $m$ and does not coincide with it.