Coordinate Geometry: Equations of Lines

Ahaan well-explained point slope form and gradient intercept form of the equation of a line. Now, we will investigate which is the best form of equation when given different information.

Given a slope and $y$-intercept.

Given a slope and a point.

Now, we learn two more forms of equation. First, the two-point form. The equation of line crossing two point $(x_1,y_1)$ and $(x_2,y_2)$,

$$\frac{y-y_1}{x-x_1}=\frac{y_2-y_1}{x_2-x_1}$$

Next, the intercept form. The equation of line crossing both $x$-axis $(a)$ and $y$-axis $(b)$,

$$\frac{x}{a}+\frac{y}{b}=1$$

Given two points.

Given two intercepts.

The slope is essentially the "steepness" of a line. The slope of a line of the form $y=ax+b$ is just $a$.

Slope is commonly remembered as "rise over run" where "rise" is the change in height, or the y-coordinates, and "run" is the change in length, or the x-coordinates. In other words, slope is the change in the y-value according to the x-value. So:

$slope=\frac{rise}{run}$

Great!

To find the slope of a line that connects or goes through two given points, $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$, the slope is found by using the expression $\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$. (It doesn't matter which point is $(x_{1}, y_{1})$ and which one is $(x_{2}, y_{2})$.)

Great job!

There are a few ways of writing the equation of a line. The first and most useful way of writing it is known as slope-intercept form.

If the slope of the line is $m$ and the line crosses the $y$-axis at the point $(0,b)$, the $y$-intercept, then the equation of the line is $$y = mx + b.$$

Clearly, the equation of the line is of the form $y = mx + 3$, where $m$ is the slope. Substituting $(4, 0)$, we get $$0 = 4m + 3 \implies m = - \frac {3}{4},$$ so the equation of the line is $y = - \frac {3}{4} x + 3$. Thus, at $x = 2000$, $y = - 1500 + 3 = \boxed {-1497}$.

Next, we talk about the point slope form of the equation of a line. If a point $(x_1,y_1)$ is crossed by a line with slope $m$, the equation of the line is $$y - y_1 = m \cdot \left( x - x_1 \right),$$ and this is the point-slope form of the equation. It can be used in the same manner as the slope-intercept form of the line.

Finally, two important properties of lines are that the slopes of lines are related in nice ways. If two lines have the same slope, they are ...

Parallel! And if they are perpendicular, how do you think the slopes are related.

Notice that if two lines, $\ell$ and $m$, are perpendicular, it can easily be proven that the product of the slopes of lines $\ell$ and $m$ is $-1$. Exercise for the reader: prove this with simple coordinates and the definition of the slope of a line.