Q.Prove that the line through the points $(2, -3)$ and $(-4, 5)$ is parallel to the line through the points $(5, 1)$ and $(-1, 9)$, and find the distance between these two parallel lines.

[!TLDR] Both lines have slope $-\dfrac43$, so they are parallel; their equations are $4x+3y+1=0$ and $4x+3y-23=0$, giving a distance of $\dfrac{24}{5}$. [!ANSWER] The lines are pa …

Slope of the line through $(2,-3)$ and $(-4,5)$: $\dfrac{5-(-3)}{-4-2}=\dfrac{8}{-6}=-\dfrac43$. Slope of the line through $(5,1)$ and $(-1,9)$: $\dfrac{9-1}{-1-5}=\dfrac{8}{-6}=-\dfrac43$. Since both slopes equal $-\dfrac43$, the two lines are parallel. Their equations (point-slope through each pair, cleared of fractions): first line through $(2,-3)$: $y+3=-\dfrac43(x-2) \implies 3y+9=-4x+8 \implies 4x+3y+1=0$ (check with $(-4,5)$: $-16+15+1=0$ ✓). Second line through $(5,1)$: $y-1=-\dfrac43(x-5) \implies 3y-3=-4x+20 \implies 4x+3y-23=0$ (check w …

Show the two slopes are equal to prove parallelism, find each line's full equation using the point-slope form, then apply the paral …

Stopping after proving the slopes are equal without going on to find the actual distance (which needs each line's full equation, no …