Question

What's the distance between equations y=2x+3 and y=2x-3

Answer 1

The distance between two parallel lines is the length of the perpendicular segment connecting the two lines.

We find the slope of the perpendicular:
$m_{perpendicular}=-\frac{1}{m}=-\frac{1}{2}$

Pick a point on $y=2x+3$ (let's go with $(0, 3)$) and find where it intersects $y=2x-3$

The perpendicular line will be $y=-\frac{1}{2}x+3$

We have a system of equations:
$y=-\frac{1}{2}x+3$
$y=2x-3$
Solve:
$2x-3=-\frac{1}{2}x+3$
⇒$\frac{5}{2}x-3=3$
⇒$\frac{5}{2}x=6$
⇒$5x=12$
⇒$x=\frac{12}{5}$
Plug into $y=2x-3$:
$y=2(\frac{12}{5})-3=\frac{24}{5}-\frac{15}{5}=\frac{9}{5}$

So our second point is $(\frac{12}{5}, \frac{9}{5})$

The distance between the points is:
$d=\sqrt{(\frac{12}{5}-0)^{2}+(\frac{9}{5}-3)^{2}}$
⇒$d=\sqrt{(\frac{12}{5})^{2}+(-\frac{6}{5})^{2}}$
⇒$d=\sqrt{\frac{144}{25}+\frac{36}{25}}$
⇒$d=\sqrt{\frac{180}{25}}$
⇒$d=\frac{\sqrt{180}}{5}$
⇒$d=\frac{6\sqrt{5}}{5}$

So the distance between the two lines is $\frac{6\sqrt{5}}{5}$

Answer 2

It is 6 as one of them is +3 and the other is -3, the difference between 3 and -3 is 6