The "point-slope" form of the equation of a straight line is:
y − y1 = m(x − x1)
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 first (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
Making sense of it
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.
Example 1:
slope "m" = 31 = 3
y − y1 = m(x − x1)
We know m, 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:
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:
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 we 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 (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 at (0, b)
and the equation becomes:
Start withy − 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
This calculator will find the equation of a line (in the slope-intercept, point-slope, and general forms) given two points or the slope and one point, with steps shown.
Related calculators: Slope Calculator, Parallel and Perpendicular Line Calculator
Solution
Your input: find the equation of a line given two points $$$P=\left(-4, 7\right)$$$ and $$$Q=\left(1, 2\right)$$$.
The slope of a line passing through the two points `P=(x_1, y_1)` and `Q=(x_2, y_2)` is given by `m=(y_2-y_1)/(x_2-x_1)`.
We have that $$$x_1=-4$$$, $$$y_1=7$$$, $$$x_2=1$$$, $$$y_2=2$$$.
Plug the given values into the formula for slope: $$$m=\frac{\left(2\right)-\left(7\right)}{\left(1\right)-\left(-4\right)}=\frac{-5}{5}=-1$$$.
Now, the y-intercept is `b=y_1-m*x_1` (or `b=y_2-m*x_2`, the result is the same).
$$$b=7-\left(-1\right) \cdot \left(-4\right)=3$$$.
Finally, the equation of the line can be written in the form `y=mx+b`.
$$$y=-x+3$$$.
Answer:
The slope of the line is $$$m=-1$$$.
The equation of the line in the slope-intercept form is $$$y=-x+3$$$.
The equation of the line in the point-slope form is $$$y - 7 = - (x + 4)$$$.
The equation of the line in the point-slope form is $$$y - 2 = - (x - 1)$$$.
The general equation of the line is $$$x + y - 3 = 0$$$.