# Point-Slope Form Of A Line

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.

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

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.