FINDING EQUATIONS OF LINES

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.
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:
$\displaystyle\text{slope} = \frac{4-(-5)}{-1-2} = \frac{9}{-3} = -3$
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!

Click here for a great line exercise, written by one of my users (it will open in a new window).
Click the ‘New line’ button to get started.
Find the $y$-intercept; find another point that lies on the intersections of the dashed grid lines, and determine the slope.
Type your equation at the bottom, and click ‘Graph my equation’ to compare.

Master the ideas from this section