Point-Slope Form: Definition & Overview

Point-Slope Form: Definition & Overview
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  • 0:00 Linear Equations
  • 1:05 Writing Linear Equations
  • 1:35 Point-Slope Form
  • 2:39 Examples
  • 6:52 Lesson Summary
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Lesson Transcript
Instructor: Jennifer Beddoe
A linear equation is an algebraic equation that, when graphed, creates a straight line. The point-slope form is one way to write a linear equation. This lesson will teach you how to write a linear equation in point-slope form and give you some examples.

Linear Equations

A linear equation is made up of one or more terms that are either constants or the product of a constant and a single variable (such as 2x). The terms of the variable must be to the single power and not squared, cubed, or more, but the equation can have more than one variable.

Linear equations have many practical uses; they're used extensively in banking and finance and can be used to help with personal finances. They also have scientific and engineering applications.

Here are some examples of linear equations:

y = 2x + 5

3m - 2n = 6

a/2 = b + 1

Here are some examples of non-linear equations:

y^2 = x + 2

√5x - 2y = 7

4/x^2 + 2y + 3 = 0

Writing Linear Equations

There are many different ways to write a linear equation, including:

  • Slope-intercept form: y = mx + b. This is the most common form of a linear equation, where m is the slope and b is the y-intercept.
  • Point-slope form: (y - y1) = m(x - x1)
  • General form: Ax + By + C = 0 (A and B cannot both be zero)

Point-Slope Form

In point-slope form (which is written like this: (y - y1) = m(x - x1)), y1 is the y value of the known point on the line, m is the slope, and x1 is the x value of the known point.

This form of a linear equation is derived from the equation for finding the slope of a line. The slope of a line is the ratio of the line's elevation to its horizontal movement, or as it's more commonly known, rise over run.

The equation for finding the slope of a line is:

m = (y2 - y1)/(x2 - x1)

You can rearrange this formula by multiplying both sides of the equations by (x2 - x1) to get:

(y2 - y1) = m(x2 - x1)

The point-slope form of a linear equation is most useful for finding a point on a line when you know the slope and one other point on the line. It can also be used to find a point on the line when you know two other points.

Examples

Let's now go over some example problems to help cement everything we've covered.

Example #1

A line goes through point (1, 3) with a slope of 2. What is the equation of the line? Start by using the point-slope formula to find the equation.

(y - y1) = m(x - x1)

y - 3 = 2(x - 1)

Then simplify.

y - 3 = 2x - 2

y = 2x + 1

Example #2

What is the equation of a line that goes through point (-2, 4) and the origin? The first step is to find the slope of the line.

m = (y2 - y1)/(x2 - x1)

It doesn't matter which point is 1 or 2; just remember to keep it consistent throughout the problem.

m = (4 - 0)/(-2 - 0)

m = 4/-2 = -2

So, the slope of the line that runs through the origin and point (-2, 4) on the coordinate plane is -2.

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