Geometry of the Plane

Slopes of Lines

Two lines $y=m_1 x +b_1$ and $y=m_2x+b_2$ are parallel (meaning they never meet), if and only if they the same slope: $m_1 = m_2$.

Two lines are perpendicular (meaning they meet at a right angle), if and only if one slope is the negative reciprocal of the other: $m_1 = -\frac{1}{m_2}$.

Example.

Consider the lines given by $y=3x-8$ and $-3x+y=42$.

They are parallel because they both have slope $3$.

Example.

Consider the lines given by $y=\frac32x-1$ and $y=-\frac23x+82$.

They are perpendicular because $\displaystyle -\frac23 = -\frac{1}{\frac32}$.

Example.

Consider the lines given by $y=2x+7$ and $2y-4x=14$.

These two equations represent the same line. We can see this easily: rewrite the second equation by dividing both sides by 2: $y-2x=7$, which is the same equation as for the first line.

Example.

Find the equation of the line perpendicular to the line $2y+4x+3=0$ and having $x$-interceptThe point where the line intercepts the $x$-axis equal to $5$.

Solution. First we write the equation $2y+4x+3=0$ in slope-intercept form: $$ y = -\frac12(4x+3) = -2x - \frac32. $$ So its slope is $-2$. This means that the slope of the perpendicular line is $-\frac{1}{-2} = \frac12$. Since $5$ is the $x$-intercept, that means that the point $(5,0)$ lies on the line, so we can write the point-slope form of the line: $$ y-0 = \frac12 (x-5), $$ Simplify this to get the equation of the line: $$ y = \frac12 x - \frac52. $$