POINT-SLOPE EQUATION OF A STRAIGHT LINE

Point-Slope Equation of a Line

The "point-slope" form of the equation of a straight line is:

y - y₁ = m(x - x₁)

Using this formula, If you know:

• one point on the line
• and the slope of the line,

you can find other points on the line.

What does it stand for?

(x1, y1) is a known point m is the slope of the line (x, y) is any other point on the line

Making Sense of It

It is based on the slope:

Slope m  =   change in y   =   y - y1 change in x x - x1

So this is the slope: and we can rearrange it like this: to get this:

So, it is just the slope formula in a different way!

Now let us see how to use it.

Example 1

slope "m"

=

3 1

= 3

y - y₁ = m(x - x₁)

We know m now, and also know that (x₁, y₁) = (3,2), and so we have:

y - 2 = 3(x - 3)

That is a perfectly good answer, but we can simplify it a little:

y - 2 = 3x - 9

y = 3x - 9 + 2

y = 3x - 7

Example 2

slope "m"

=

3 -1

= -3

y - y₁ = m(x - x₁)

We can pick any point for (x₁, y₁), so let's choose (0,0), and so we have:

y - 0 = -3(x - 0)

Which can be simplified to:

y = -3x

Example 3: Vertical Line

What is the equation for a vertical line?
The slope is undefined!

In fact, this is a special case, and you use a different equation, like this:

x = 1.5

Every point on the line has x coordinate 1.5,
that’s why its equation is x = 1.5

What About y = mx + b ?

You may already be familiar with the "y=mx+b" form.

It is the same equation, in a different form!

The "b" value (called the y-intercept) is where the line crosses the y-axis.

So point (x₁, y₁) is actually at (0, b)

and the equation becomes:

Start with	y - y1 = m(x - x1)
(x1, y1) is actually (0, b):	y - b = m(x - 0)
Which is:	y - b = mx
Put b on other side:	y = mx + b

And that is called the "slope-intercept" form of the equation of a line.