What is the point-slope form of a line that has a slope of 3 and passes through point (1, 4)?

We can also find the equation of a line when given the slope and any point (not the y-intercept), and there are two methods to do so. The following video will use a single example to show how to use both methods to find the equation of a line with a given slope and single point.

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What does it stand for?
Now let us see how to use it.
What About y = mx + b ?

Video Source (09:40 mins) | Transcript

These are the two methods to finding the equation of a line when given a point and the slope:

Substitution method = plug in the slope and the (x, y) point values into y = mx + b, then solve for b. Use the m given in the problem, and the b that was just solved for, to create the equation y = mx + b.
Point-slope form = ${\text{y}} {-} {\text{y}}_1 = {\text{m}}({\text{x}}-{\text{x}}_1)$, where $({\text{x}}_1, {\text{y}}_1)$ is the point given and m is the slope given. The 'x' and the 'y' stay as variables.

Additional Resources

Find the equation of the line that passes through the point (1, 4) and has a slope of 12.

Find the equation of the line that passes through the point (1, 4) and has a slope of 2.

Find the equation of the line that passes through the point (27, 4) and has a slope of $\frac{-2}{9}$.

Find the equation of the line that passes through the point $(-11, 2)$ and has a slope of $\frac{-5}{11}$.

Find the equation of the line that passes through the point (10, 6) and has a slope of $\frac{1}{5}$. What is the y-intercept of the line?

Find the equation of the line that passes through the point (3, 29) and has a slope of 6. What is the y-intercept of the line?

The equation is useful when we know:

one point on the line: (x1, y1)
and the slope of the line: m,

and want to find other points on the line.

Have a play with it (move the point, try different slopes):

Now let's discover more.

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

It is based on the slope:

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

Starting with the slope:

we 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.

slope "m" = 31 = 3

y − y1 = m(x − x1)

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

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

y − 2 = 3x − 9

y = 3x − 9 + 2

y = 3x − 7

m = −3 1 = −3

y − y1 = m(x − x1)

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

y − 0 = −3(x − 0)

Which can be simplified to:

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

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

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 (called the slope-intercept form of the equation of a line).

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 (0, b)

and the equation becomes:

Start withy − y1 = m(x − x1)

(x1, y1) is (0, b):y − b = m(x − 0)

Which is:y − b = mx

Put b on other side:y = mx + b

519, 521, 522, 1160, 1161, 1162, 2074, 2075, 9027, 9028, 9029

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The point-slope form calculator will show you how to find the equation of a line from a point on that line and the line's slope. Soon, you will know what is point-slope form equation, and learn how is it different from the slope-intercept form equation. We also came up with two exercises, and we'll explain how to solve them in the last paragraph.

Let's start with the basics. What is the slope? The slope, also known as the gradient, is the marker of a line's steepness. If it's positive, it means the line rises. If it's negative - the line decreases. If it's equal to zero, the line is horizontal.

You can find the slope between two points by estimating rise over run - the difference in height over a distance between two points.

So, slope formula is:

m = change in y / change in x = (y - y₁) / (x - x₁)

The point-slope form equation is a rearranged slope equation.

To find the gradient of non-linear functions, you can use the average rate of change calculator.

🙋 For more information go to the slope calculator.

There is more than one way to form an equation of a straight line. Point-slope form is a form of a linear equation, where there are three characteristic numbers - two coordinates of a point on the line, and the slope of the line. The point slope form equation is:

$y−y1=m⋅(x−x1)\small y - y_1 = m \cdot (x - x_1)$

,where:

$x1,x2\small x_1, x_2$ are the coordinates of a point, and
$m\small m$ is the slope.

Do you see the similarity to the slope formula? What you might not know is that it's not the only way to form a line equation. The more popular is the slope intercept form:

$y=m⋅x+b\small y = m \cdot x + b$

,where:

$m\small m$ is the slope; and
$b\small b$ is the intercept of the y-axis.

The truth is that this is nothing else than a more precise point-slope form. A straight line intercepts the y-axis in a point (0, b). If you choose this point - (0, b), as a point that you want to use in the point-slope form of the equation, you will get:

$y−b=m⋅(x−0)\small y - b = m \cdot (x - 0)$ , which is the same as $y=m⋅x+b\small y = m \cdot x + b$ .

In the two graphs below, you can see the same function, only described with two different forms of a linear equation:

To learn how to find the x-intercept and y-intercept of a line, visit our x- and y-intercept calculator.

Let's have a look at two exercises, to understand the topic more clearly.

The slope of a line is 2. It passes through point A(2, -3). What is the general equation of the line?

Identify the point coordinates:
Identify the slope:
Input the values into the point slope form formula:
- $y−y1=m(x−x1)\small y - y_1 = m (x - x_1)$
- $y−(−3)=2(x−2)\small y - (-3) = 2(x - 2)$
Simplify to get the general equation:
- $y=2x−4−3\small y = 2x - 4 -3$
- $0=2x−y−7\small 0 = 2x - y - 7$
And you have the answer. Now, you can check your result with our point-slope form calculator.

Let's solve an exercise with a more relatable subject.

Let's say you got a puppy. When you got him he was 14 pounds. It grew 0.2 pounds every day, and after 30 days, he was 20 pounds. Find the general equation of the puppy's growth.