Finding Equations of Lines

Want some practice with related concepts first?

Every non-vertical line in the coordinate plane can be described by an equation of the form $\,y = mx + b\,,$ where:

$\,m\,$ is the slope of the line
$\,b\,$ is where the line crosses the $\,y$-axis

The equation $\,y = mx + b\,$ is called the slope-intercept form of the line.

Two different points uniquely determine a line. One point and a slope also uniquely determine a line. This web exercise gives you practice writing the equation of the line in these two situations.

Example (Known Point, Known Slope)

Question: Find the equation of the line with slope $\,3\,$ that passes through the point $\,(-1,5)\,.$ Write the equation in $\,y = mx + b\,$ form.

Solution:

$y = mx + b$	A line with slope $\,3\,$ isn't vertical, so it can be described by an equation of this form.
$y = 3x + b$	Substitute the known slope, $\,3\,,$ in for $\,m\,.$ Next, we must find $\,b\,.$
$5 = 3(-1) + b$	Since $\,(-1,5)\,$ lies on the line, substitution of $\,-1\,$ for $\,x\,$ and $\,5\,$ for $\,y\,$ makes the equation true.
$5 = -3 + b$	simplify
$\,b = 8\,$	add $\,3\,$ to both sides; write in the conventional way
$y = 3x + 8$	substitute the now-known value of $\,b\,$ into the equation

Thus, the line with slope $\,3\,$ that passes through $\,(-1,5)\,$ is described by the equation $\,y = 3x + 8\,.$

Make sure you understand what this means! Let $\,\ell\,$ denote the line with slope $\,3\,$ that passes through the point $\,(-1,5)\,.$ Every point that lies on $\,\ell\,$ has coordinates that make the equation $\,y = 3x + 8\,$ true. Every point that doesn't lie on $\,\ell\,$ has coordinates that make the equation $\,y = 3x + 8\,$ false.

Head up to wolframalpha.com and type in:

y = 3x + 8, x = -1, y = 5

(Cut-and-paste, if you want.) You'll see a graph of the line, with the given point indicated by crosshairs. By adding in an additional set of crosshairs, you can see that going up $\,3\,$ and to the right $\,1\,$ brings you to another point on the line:

y = 3x + 8, x = -1, y = 5, x = 0, y = 8

Example (Two Known Points)

Question: Find the equation of the line through the points $\,(2,-5)\,$ and $\,(-1,4)\,.$ Write the equation in $\,y = mx + b\,$ form.

Solution: First, use the slope formula to compute the slope: $$ \cssId{s51}{\text{slope}} \cssId{s52}{= \frac{4-(-5)}{-1-2}} \cssId{s53}{= \frac{9}{-3}} \cssId{s54}{= -3} $$ Then, continue as in the previous example:

$y = mx + b$	start with slope-intercept form
$y = -3x + b$	substitute the now-known slope, $\,-3\,,$ in for $\,m\,$
$4 = -3(-1) + b$	Which point should you use? It doesn't matter! In general, try to choose the simplest numbers to work with.
$4 = 3 + b$	simplify
$\,b = 1\,$	subtract $\,3\,$ from both sides; write in the conventional way
$y = -3x + 1$	substitute the now-known value of $\,b\,$ into the equation

You might want to check that the two points do indeed lie on the line:

$-5\ \overset{\text{?}}{ = } -3(2) + 1\,$ Check!
$4\ \overset{\text{?}}{ = } -3(-1) + 1\,$ Check!