Finding The Equation of a Line

An example of finding the equation of a line

In practice you will often be asked to find the equation of a line that is passing through a given point and is parallel to another line.  A line is parallel to another line if they have the same slope (m).  I think the best way to show how to do this would be to do an example problem.


Find the equation of the line L passing through the point (2, 3) and parallel to the line y = -3x + 5

The general equation for a line is y = mx + b

where m = the slope (rise over the run), x and y are your variables, and b is the point where the line intersects the y-axis.

Because the line we are finding is parallel to the one given, we know they have the same slope, which is -3 here.  So far we know this about our new line:

y = -3x + b

the slope of the line is defined as the change in distance y divided by the change in distance x on a coordinate plane, it is written like so, \frac{\Delta y}{\Delta x} .  So what can we do with this equation?

We know that our line is going to pass through the point (2 , 3), and \frac{\Delta y}{\Delta x} = \frac{y - y_{0}}{x - x_{0}} = -3 where x_{0} and y_{0} are our known points, x = 2 and y = 3 .  This relation works because we are dividing the difference in distance between any point y and our known point, by the difference in distance between any point x and our known point, giving us an equation in terms of x and y.

if we simplify \frac{y - y_{0}}{x - x_{0}} = -3 we get y-3 = -3x + 6

We want this equation in the form y = mx + b so move the “-3” over and our final equation is

y = -3x + 9

Graphing the equation of the line

Let’s make sure we’re right.  The equation should have the same slope (m) as the equation we were given, which is -3 and this checks out.  We also know that we used our given point to and the slope formula to reach this conclusion, so all signs point to this being pretty darn accurate.  Go ahead and graph it to make sure.


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