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Finding The Equation Of A Line, Given Two Points

This video explains how to find the equation of a line when given 2 points.

Find the slope using the slope formula,  m = \frac{y_2 - y_1}{x_2 - x_1} .
After finding the slope, plug in the value of your points into either slope-intercept form, or point-slope form, to get the equation of the line.

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Video-Lesson Transcript

Let’s find an equation of a line, given two points.

How do we do this?

We can solve this by using the slope intercept form:

y = mx + b

or by using point-slope formula:

y - y_1 = m (x - x_1)

But in both cases we need to know what the slope (m) is.

So the first step is to find the slope.

The formula for finding the slope is

m = \dfrac{y_2 - y_1}{x_2 - x_1}

For example:

Find the equation of a line with given points 2, 4 and 4, 5

First, let’s find the slope.

Let’s label the given:

x_1 = 2, y_1 = 4
x_2 = 4, y_2 = 5
m = \dfrac{y_2 - y_1}{x_2 - x_1}
m = \dfrac{5 - 4}{4 - 2}
m = \dfrac{1}{2}

So now, we can find the equation of a line using the slope intercept form or the point slope form.

Let’s do the slope intercept form first.

y = mx + b
y = \dfrac{1}{2}x + b

Now, you can substitute any of the paired coordinates but not both.

I’m going to use the second pair of coordinates x_2 = 4, y_2 = 5

5 = \dfrac{1}{2} (4) + b

Now, let’s solve for b

5 = 2 + b
5 - 2 = 2 - 2 + b
3 = b

Now, we know what the y-intercept is and the slope.

So the equation y = mx + b becomes

y = \dfrac{1}{2}x + 3

Finding the Equation of a Line 2

Now, let’s solve using the point-slope form.

y - y_1 = m (x - x_1)

Here, we can choose which pair to use as coordinates.

So let me choose the first set of coordinates x_1 = 2, y_1 = 4

Also, we already know what the slope is m = \dfrac{1}{2}

Now, let’s solve

y - 4 = \dfrac{1}{2} (x - 2)
y - 4 = \dfrac{1}{2}x - 1

Let’s get y by itself by adding 4 on both sides.

y - 4 + 4 = \dfrac{1}{2}x - 1 + 4

And our answer is

y = \dfrac{1}{2}x + 3

which is the same as the first one.