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.
What is a slope?
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.
What is the point-slope form?
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),
y1are the coordinates of a point, and
mis 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,
mis the slope; and
bis 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), which is the same as
y = m * x + b.
In the two graphs below, you can see the same function, only described with two different forms of a linear equation:
How to find the equation of a line with slope and coordinates of a point?
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:
x1 = 2,
y1 = -3.
- Identify the slope:
m = 2
- Input the values into the point slope form formula:
y - y1 = m (x - x1)
y - (-3) = 2(x - 2)
- Simplify to get the general equation:
y = 2x - 4 -3
0 = 2x - y - 7And 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.
- The slope is the change of weight per day:
m = 0.2
- The characteristic point is 20 pounds on 30th day:
(x1, y1) = (30, 20)
- Now, input the values into the point-slope formula:
y - 20 = 0.2 * (x - 30)
- Simplify the equation to get the general equation:
0 = 0.2x - y + 14
|💡 If you need to find a different point on your line click on the |
And here you have it! We hope you enjoyed our point-slope form calculator! Before you go, check out more of our geometry calculators!