Find the slope intercept equation of a line (y=mx+b or y=mx+c) from two points with this slope intercept form calculator.
Slope Intercept Form Equation: y = mx + b, or sometimes y = mx + c,
m = slope (the amount of rise over run of the line)
b= y-axis intercept ( where the line crosses over the y-axis)
To calculate the slope intercept form equation from two coordinates
(x1,y1) and (x2,y2):
Step 1: Calculate the slope (y2 - y1) / (x2 - x1)
Step 2: Calculate where the line intersects with the y-axis by
entering one of the coordinates into this equation: y - mx = b
Example:
To calculate the slope-intercept equation for a line that includes
the two points ( 7, 4) and (1, 1).
Step 1: slope (m) = (1 - 4) / (1 - 7) = -3 / -6
slope (m) = -3/-6 = 1/2
Step 2: Using one of the original coordinates (7, 4) we find
the
y-axis intercept (b) using the formula: y - mx = b
y=4, m=1/2, x =7
y - mx = b
b= .5
The slope intercept form for this line is y = .5x + .5
This line crosses the y-axis at .5 and has a slope of .5,
so this line rises one unit along the y-axis for every 2 units
it moves along the x-axis.
So, where would you ever use this? Here's an article on ways to use the Slope Intercept Form in Real Life.
You can find an equation of a straight line given two points laying on that line. However, there exist different forms for a line equation. Here you can find two calculators for an equation of a line:
first calculator finds the line equation in slope-intercept form, that is,
It also outputs slope and intercept parameters and displays the line on a graph.- second calculator
finds the line equation in parametric form, that is,
It also outputs a direction vector and displays line and direction vector on a graph.
Also, the text and formulas below the calculators describe how to find the equation of a line from two points manually.
Slope-intercept line equation from two points
First Point
Second point
Calculation precision
Digits after the decimal point: 2
Parametric line equation from two points
First Point
Second point
Calculation precision
Digits after the decimal point: 2
How to find the equation of a line in slope-intercept form
Let's find slope-intercept form of a line equation from the two known points and .
We need to find slope a and intercept b.
For two known points we have two equations in respect to
a and b
Let's subtract the first from the second
And from there
Note that b can be expressed like this
So, once we have a, it is easy to calculate b simply by plugging or
to the expression above.
Finally, we use the calculated a and b to write the result as
Equation of a vertical line
Note that in the case of a vertical line, the slope and the intercept are undefined because the line runs parallel to the y-axis. The line equation, in this case, becomes
Equation of a horizontal line
Note that in the case of a horizontal line, the slope is zero and the intercept is equal to the y-coordinate of points because the line runs parallel to the x-axis. The line equation, in this case, becomes
How to find the slope-intercept equation of a line example
Problem: Find the equation
of a line in the slope-intercept form given points (-1, 1) and (2, 4)
Solution:
- Calculate the slope a:
- Calculate the intercept b using coordinates of either point. Here we use the
coordinates (-1, 1):
- Write the final line equation (we omit the slope, because it equals one):
And here is how you should enter this problem into the calculator above: slope-intercept line equation example
Parametric line equations
Let's find out parametric form of a line equation from the two known points and .
We need to find components of the direction
vector also known as displacement vector.
This vector quantifies the distance and direction of an imaginary motion along a straight line from the first point to the second point.
Once we have
direction vector from to , our parametric equations will be
Note that if , then
and if , then
Equation of a vertical line
Note that in the case of a vertical line, the horizontal displacement is zero because the line runs parallel to the y-axis. The line equations,
in this case, become
Equation of a horizontal line
Note that in the case of a horizontal line, the vertical displacement is zero because the line runs parallel to the x-axis. The line equations, in this case, become
How to find the parametric equation of a line example
Problem: Find the equation of a line in the parametric form given points (-1, 1) and (2, 4)
Solution:
- Calculate the displacement vector:
- Write the final line equations: