Geometry of the Plane

Lines

Lines can be written in slope-intercept form $$ y=mx+b $$ where $m$ is the slope of the line and $b$ is the $y$-interceptThe point where the line intercepts the $y$-axis or in point-slope form $$ y - y_1 = m(x-x_1) $$ where $m$ is the slope of the line and $(x_1,y_1)$ is a point on the line.
Note that this last formula is equivalent to the formula for the slope of a line $$ m = \frac{y_2-y_1}{x_2-x_1}, $$ where $(x_1,y_1)$ and $(x_2,y_2)$ are two points on the line.

All these forms are equivalent. To find the equation of a line, list the values given, substitute into one of these equations, and solve for the missing parameters.

Example.

What is the equation of a line with slope $3$ and $y$-intercept $(0,-1)$?

Solution. Using the second equation for a line, we obtain: $y=3x-1$.

Example.

What is the equation of a line that goes through $(3,-2)$ and $(5,0)$?

Solution. We have two points on the line, so we use the point-slope form.
Use the first point $(x_1,y_1)=(3,-2)$.
We need to find $m$: use the other point with the formula $$ m = \frac{y_2-y_1}{x_2-x_1} = \frac{0-(-2)}{5-3} = \frac22 = 1, $$ so the equation of this line is $$ y -(-2)=1\cdot(x-3) \quad \text{ or } \quad y = x-5 $$