# Point of Intersection: Definition & Formula

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• 0:02 Point of Intersection
• 0:56 Finding a Point Graphically
• 2:14 Finding a Point Algebraically
• 4:14 Lesson Summary
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Lesson Transcript
Instructor
Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

Expert Contributor
Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

In this lesson, we'll learn what the point of intersection is and how to find it graphically and algebraically. This concept applies to lots of other areas of study, including business, finance, construction and physics.

## Point of Intersection

Have you ever been out driving and noticed a traffic sign that looks like this?

This is an intersection traffic sign, and it indicates that you are coming up on a point at which two roads meet. Notice that in the intersection traffic sign, two lines cross and meet each other in the middle. This is their point of intersection. In mathematics, we call the place where two lines or curves meet their point of intersection.

The point of intersection of two curves is significant in that it is the point where the two curves take on the same value. This can be useful in a variety of applications. For instance, say we were working with an equation representing the revenue of a company and an equation representing the cost of a company. The point of intersection of the curves corresponding to these two equations would be the point where revenue is equal to cost; the breakeven point for the company.

## Finding a Point Graphically

One way of finding the point of intersection of two lines is to graph both and see where they visually intersect. For example, suppose we were trying to find the point of intersection of the lines y = x + 2 and y = 2x + 1. We would first graph both of the lines on the same graph and then identify the point at which they intersect.

Here, we see that the two lines intersect at the point (1,3). The downfall to finding the point of intersection graphically is that there is a lot of room for error. For instance, consider the two lines y = (20 / 11)x - (16 / 11) and y = (20 / 21)x + (4 / 21).

Notice how it appears as though they intersect at the point (2, 2)? In actuality, they intersect at the point (1.9, 2). With this example, it's easy to see why graphing isn't always the best way to find a point of intersection. However, it is an incredibly useful tool to check a point of intersection that you have already found. So how do we accurately find a point of intersection that can be checked using a graph? The answer lies in finding this point algebraically.

## Finding a Point Algebraically

When we are given two equations of lines, we can find the point of intersection of these lines algebraically using the following steps:

1. Solve each of the equations for y.
2. Set the two expressions for y equal to each other and solve for x. This is your x-value of the point of intersection.
3. Plug the value of x into either one of the original equations and solve for y. This is your y-value for the point of intersection.

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## Point of Intersection Map Application

### Review

• A point of intersection is a point where two lines or curves meet.
• We can find a point of intersection graphically by graphing the curves on the same graph and identifying their points of intersection.
• We can find a point of intersection algebraically using the following steps:
• Solve each equation for one of the variables, call it y.
• Set the expressions for y found in the first step equal to one another, and solve for the other variable, call it x. This is the x-value of the point of intersection.
• Plug the x-value of the point of intersection back into either of the original equations, and solve for y. This is the y-variable for the point of intersection.

### Problem

Sandy has to run errands today in a small town she is staying in for work. She needs to visit the post office, the grocery store, the dog park, the mall, and the hardware store. She is unfamiliar with the area, but she has the following information.

• If the roads in the town were placed on the same xy-plane, then their equations would be as follows:
• Oak Street: y = x + 4
• Elm Street: y = -3x - 6
• Willow Street: y = 5
• Pine Street: x = 4
• The post office is at the intersection of Oak and Elm.
• The grocery store is at the intersection of Oak and Willow.
• The dog park is at the intersection of Elm and Willow.
• The mall is at the intersection of Elm and Pine.
• The hardware store is at the intersection of Oak and Pine.

Given this information, draw a map for Sandy with the roads of the town on it. Find the points where the post office, the grocery store, the dog park, the mall, and the hardware store are on the map. Plot those points and label them with their point on the graph.

### Solution:

• The post office is at the point (-2.5,1.5).
• The grocery store is at the point (1,5).
• The dog park is at the point (-3.7,5).
• The mall is at the point (4,-18).
• The hardware store is at the point (4,8).

To create the map, we graph the equations of the roads given, and then plot the points of the different places Sandy needs to go.

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