Point Slope Form Calculator

Our point-slope form calculator is a handy tool for finding the equation of a line from a point on that line and the line's slope. Our calculator shows the equation in point-slope form and slope intercept form.

In the sections below, you are going to learn more about slope, point-slope form and slope intercept form.

What is a Slope?

The slope is a measure of how steep a line is. If the slope is positive, it means the line goes upward from left to right. If the slope is negative, the line goes downward from left to right. If the slope is zero, the line is perfectly flat or horizontal.

You can find the slope between two points by calculating the "rise over run." The rise is the vertical change in height, and the run is the horizontal distance between the two points.

The formula for slope is:
Slope = Change in y / Change in x = (y₂ - y₁) / (x₂ - x₁)

The point-slope form is a way to write the equation of a line using a point on the line and the slope.

What is the Point-Slope Form?

The point-slope form is an equation that describes a line using a point the line passes through and the slope of the line. The equation is:

y - y1 = m(x - x1)

Where:
- (x₁, y₁) are the coordinates of a point on the line
- m is the slope of the line

This form is similar to the slope formula. Another common way to write a line equation is the slope-intercept form:

y = mx + b

Where:
- m is the slope
- b is the y-intercept (where the line crosses the y-axis)

The slope-intercept form is actually just a more specific case of the point-slope form. If you choose the point (0, b) in the point-slope form, you get:

y - b = m(x - 0)
Which simplifies to:
y = mx + b

FAQ

How do I find the equation of a line using slope and a point?

Follow these steps:
1. Identify the slope (m) and the point coordinates (x₁, y₁)
2. Substitute into the point-slope form: y - y1 = m(x - x1)
3. Simplify to get the line equation

When would I use point-slope vs slope-intercept form?

Use point-slope when you know a point and the slope. Use slope-intercept when you know the slope and y-intercept.

Examples:

1. A line has slope 3 and passes through (2, -5). Find the equation.
* Slope m = 3
* Point is (2, -5), so x₁ = 2, y₁ = -5
* Substitute into point-slope: y - (-5) = 3(x - 2)
* Simplify: y = 3x - 6 - 5
y = 3x - 11

2. A puppy weighs 10 lbs initially. It gains 0.3 lbs per day. After 20 days, it weighs 16 lbs.
* Weight is y, days is x
* Slope m = 0.3 (lbs gained per day)
* Point is (20, 16), so x₁ = 20, y₁ = 16
* Substitute: y - 16 = 0.3(x - 20)
y = 0.3x - 6 + 16
y = 0.3x + 10

So the puppy's growth follows y = 0.3x + 10

How do I convert point-slope to slope-intercept form?

Follow these steps:
1. Start with the point-slope form: y - y1 = m(x - x1)
2. Expand the right side: y - y1 = mx - mx1
3. Add y₁ to both sides: y = mx - mx1 + y1
4. This is slope-intercept form! The slope is m, and the y-intercept is -mx₁ + y₁.

How do I calculate the y-intercept given point-slope form?

The y-intercept of y - y1 = m(x - x1) is y₁ - mx₁. For example, if the equation is y - 1 = 2(x - 3), then the y-intercept is 1 - 2(3) = -5.

What is the point-slope formula with zero slope?

If the slope m = 0, the point-slope form becomes y - y1 = 0. This represents a horizontal line at y = y₁.

Can point-slope form be the same as slope-intercept form?

Yes! Consider the slope-intercept form y = mx + b. If we choose the point (0, b), the point-slope form becomes:
y - b = m(x - 0)
Which is equivalent to y = mx + b after rearranging.