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Point-Slope Form Of A Line

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The point slope form is written in the format:

y - y_1 = m(x - x_1)

Here’s an example:

 m = \frac{1}{2}
x_1 = 6
y_1 = -3

In point slope format, this would be written as:

 y - (6) = \frac{1}{2}(x - (-3))
 y - 6 = \frac{1}{2}(x+3)

Video-Lesson Transcript

Let’s go over point slope form.

The actual equation is:

y - y_1 = m (x - x_1)

where m is the slope of the line, x_1 and y_1 are the coordinate of the line.

For example, we have:

m = \dfrac{5}{2}, (2, 7)

Let’s solve by substituting the values

y - y_1 = m (x - x_1)
y - 7 = \dfrac{5}{2} (x - 2)

That’s it.

Let’s have another one.

m = \dfrac{1}{2}, (-3, 6)
y - y_1 = m (x - x_1)
y - 6 = \dfrac{1}{2} (x + 3)
Point-Slope Form

Let’s discuss this further.

We already know the formula for slope is:

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

Let’s say we write this as fraction

\dfrac{m}{1} = \dfrac{y_2 - y_1}{x_2 - x_1}

Now, let’s cross multiply. It will be

m (x_2 - x_1) = y_2 - y_1

Then instead of writing x_2 and y_2, let’s just write them as x and y.

Now, our equation will look like this

m (x - x_1) = y - y_1

This is already the point-slope form, so let’s just rewrite this as

y - y_1 = m (x - x_1)

So the point-slope form is just derived from the slope formula.

From the point-slope form, we can find the equation of a line in a slope intercept.

For example:

m = 2, (3, -1)
y - y_1 = m (x - x_1)
y + 1 = 2 (x -3)

The equation of a line in a slope intercept is:

y = mx + b

Let’s continue solving by distributing 2 in the parenthesis.

y + 1 = 2x - 6

Then isolate y

y + 1 - 1 = 2x - 6 - 1
y = 2x - 7

So now, our answer is the equation of a line in a slope intercept form which we dervied from the point slope form.