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Calculus: Help and Review13 chapters | 148 lessons

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Lesson Transcript

Instructor:
*Laura Pennington*

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

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.

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.

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* = 2*x* + 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.

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

- Solve each of the equations for
*y*. - Set the two expressions for
*y*equal to each other and solve for*x*. This is your*x*-value of the point of intersection. - 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.

Let's put these steps into action by finding the point of intersection of our last example, algebraically this time. We want to find the point of intersection of the two lines *y* = (20 / 11)*x* - (16 / 11) and *y* = (20 / 21)*x* + (4 / 21). We can eliminate the first step since both of the equations have already been solved for *y*. The next step is to set the two expressions for *y* equal to each other and then solve for *x*.

(20/11)x - (16/11) = (20/21)x + (4/21) |
Multiply both sides by 231, eliminating fractions. |

420x - 336 = 220x + 44 |
Add 336 to both sides. |

420x = 220x + 380 |
Subtract 220x from both sides. |

200x = 380 |
Divide both sides by 200. |

x = 380/200 = 1.9 |
1.9 is the x-value of the point of intersection. |

Now we plug *x* = 1.9 into either one of the original equations to find the *y*-value for the point of intersection. Let's use *y* = (20 / 11)*x* - (16 / 11), but again, we can use either one.

y = (20 / 11)x - (16 / 11) |
Plug in x = 1.9 for x. |

y = (20 /11)(1.9) - (16/11) |
Simplify. |

y = (38 / 11) - (16 /11) |
Simplify further. |

y = 22 / 11 = 2 |
2 is the y-value of the point of intersection. |

Thus, our point of intersection is (1.9, 2).

The **point of intersection** of two lines or curves is the place where the two lines or curves meet. To find the point of intersection algebraically, solve each equation for *y*, set the two expressions for *y* equal to each other, solve for *x*, and plug the value of *x* into either of the original equations to find the corresponding *y*-value. The values of *x* and *y* are the *x*- and *y*-values of the point of intersection. You can check this point of intersection by graphing the two equations and verifying that they do, in fact, intersect at this point. Being able to find this point is extremely useful and a great tool to have in your mathematical toolbox.

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Calculus: Help and Review13 chapters | 148 lessons

- What is a Function: Basics and Key Terms 7:57
- Graphing Basic Functions 8:01
- Compounding Functions and Graphing Functions of Functions 7:47
- Understanding and Graphing the Inverse Function 7:31
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