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How to Find Equations of Lines With Given Slopes & Points

Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

This lesson will demonstrate how to find the equation of a line knowing only the slope and a point on the line or two points on the line. We will see why this is a useful tool and look at an example on how to apply it.

Slope of a Line

Have you ever been walking up a steep hill and wondered just how steep it was? In other words, how far up are you going for every unit you are going horizontally?


Slope
pointslope1


This rate at which your vertical distance is changing with respect to your horizontal distance is called the slope of the hill. In mathematics, the slope of a line determines how steep the line is. It is the rate at which y is changing with respect to x. When we have two points on a line, (x 1, y 1) and (x 2, y 2), we can calculate the slope using the following formula, where m is the slope:


pointslope2


In this lesson, we're going to see how we can find the equation of a line with just knowing the slope of the line and a point on that line, or even by just knowing two points. Let's get started!

Point-Slope Form of a Line

When it comes to the equation of a line, we have a couple of special forms of the equation. One of those is the point-slope form of a line. The point-slope form of a line is a formula that is useful when we know the slope, m, of a line and a point on that line, (x 1, y 1). We can plug the slope and the point into the following point-slope form of the line to find an equation of the line.

y - y 1 = m(x - x 1)

Let's consider an example of this. Suppose we have a line with slope m = 4, and it contains the point (x 1, y 1) = (5,2). We can plug these values into our point-slope form of our line, and we have an equation of the line.

y - 2 = 4(x - 5)

We can simplify this by distributing the 4 and then adding 2 to both sides.


pointslope3


We see that the equation of our line is y = 4x - 18. Pretty cool that we figured that out with just the slope of the line and a point on the line!

Now, suppose I told you I have a line with the points (5,2) and (1,14). Any ideas on how we could find the equation of the line? We have a point on the line. Either one will do, but we need to have the slope as well. Ah-ha! We can use our slope formula to find the slope of the line and then use that and one of the points to plug into our point-slope form of the line.

First, we need to find our slope.


pointslope4


We see our slope is m = -3. Now we just need to plug that slope and either one of the points on the line into our point-slope form of the line and simplify! Let's use (5,2), but you will get the same result with (1,14).


pointslope5


We see that the equation of our line is y = -3x + 17. All that from just knowing two points on the line!

Why is This Useful?

Once we have an equation for a line, we can analyze the line and what it is modeling more easily. For example, suppose you are setting aside $50 per week to save up for a bike that costs $450. You are two weeks in. After the first week, you have $50, and after the second week, you have $100. You want to know how many weeks it will take to get to $450.

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