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The *intersection *is the place (x,y) where two functions cross each other on a graph. You may want to find the intersection of two lines for many reasons. Perhaps the most important reason is that the intersection of two graphs is the solution to a series of equations (which is much easier than solving systems of equations algebraically!).

**Three ways to find the intersection of two lines (click to skip to that section):**

## Find the Intersection of two lines on a TI-89

An intersection is where two (or more) functions meet on a graph. Finding an intersection is one way to solve a *system of equations*; the point where the two graphs cross each other (intersect) is the solution to the system. Using a TI 89 to find the intersection is much faster than the hand method and is no harder than pressing a few buttons.

## How to Use Intersection and Trace

## Intersection Feature

**Example problem**: find the intersection of two functions:

f(x) = x^{2} + 3x + 7

f(x) = x^{2} + 5x + 9

Step 1: Press **HOME**.

Step 2: Press the **diamond key** and then **F1 **to enter into the *y=editor*. Next, press the **CLEAR **button if there are any values in the y1 slot and then press **ENTER **to go down to the input line.

Step 3: Enter the first function/equation. For this example, press **x ^ 2 + 3 x + 7**.

Step 4: Press **ENTER **to enter the function into the “y1 =” slot.

Step 5: Enter the second function. Press** x ^ 2 + 5 x + 9**.

Step 6: Press **ENTER **. This puts the second function into the “y2 =” slot.

Step 7: Press **HOME**.

Step 8: View the graph by pressing the **diamond key** and then **F3 **. You will see that the two graphs intersect. *Note*: If you don’t see a graph, press **F2 **and then press **6**.

Step 9: Press **F5 **and then **5** to select “Intersection.”

Step 10: When you are asked “1st curve?” press **ENTER**.

Step 11: When you are asked “2nd curve?” press **ENTER**.

Step 12: For the lower bound, press the left arrow, moving the arrow to the left of the intersection. Then press **ENTER**.

Step 13: For the upper bound, arrow to the right of the intersection and press **ENTER**.

The TI-89 will give you an “x” value of -1 and a “y” value of 5. The intersection of these two graphs is (-1,5).

That’s it! You’re done!

## Trace Feature

The trace feature can come in handy to find your place on the graph.

Step 1: Press **F3** for the *Trace *feature.

Step 2: Press the left arrow or the right arrow to trace along the graph. Change which graph you trace along by pressing the up or down arrows.

Step 3: To see a particular value for the function, press the desired value and then press **ENTER**. For example to see what y equals for an x-input of 4, press 4 and then press **ENTER**.

That’s it! You’re done!

## How to Find the Intersection of Two Lines Algebraically

**Example problem:** Find the intersection for the linear functions

3x + 2 and 2x -1.

Step 1: Set the equations equal to each other.

3x + 2 = 2x – 1

Step 2: **Solve for x **to find the x-intersection.

- 3x + 2 = 2x – 1
- Subtract 2 from each side: 3x = 2x – 1 – 2
- Add 2x to each side: 3x – 2x = -1 – 2
- Simplify: x = -3

The x-intersection is -3.

Step 3: Use the value you found in Step 2 to find y. Take one of the original equations (we’ll use 3x + 2) and plug in the x-value:

y = 3x + 2

y = 3(-3) + 2 = -7

The intersection for the two lines is (-3, -7)

## Free Online Calculator

If you perform the steps by hand, you can use an online graphing calculator to check your work. This free online calculator works much in the same way as the TI-89 (albeit with stripped down features. It’s simple to use— even if you’ve never used a graphing calculator before.

Step 1: Go to *this URL on HRW.com* (safe site: it’s owned by a major textbook publisher, Houghton Mifflin Harcourt).

Step 2: Input your two equations. The following image shows what the calculator looks like after the equations have been entered:

Step 3: Click “GRAPH”. It’s the orange button to the right.

You can see the intersection of the two lines at the bottom left of the image. Next, we want to find out exactly what the coordinates of those lines are.

Step 4: Choose the Intersection Tab (towards the top of the page).

Step 5: Click in the check boxes next to your equations.

Step 6: Click the orange “Find intersection points” button. The intersection will show up in the box. For this set of equations, the intersection shows up at [-3,-7], which is what we expected from our graph.

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