# How to Find Intersection of Two Lines: Algebraically & TI-89

You may want to find intersection of two lines for many reasons in calculus. The most basic use is that the intersection of two graphs is the solution to a series of equations (much easier than solving systems of equations algebraically!). In calculus, finding intersections is a fundamental part of performing integral calculus.

Two ways to find the intersection of two lines:

1. Use “intersection” on a TI-89 graphing calculator,
2. Find intersections algebraically with substitution.

## Find 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 of equations. Using a TI 89 to find the intersection is much faster than the hand method and is no harder than pressing a few buttons.

## Intersection Feature

Sample problem: find the intersection of two functions:
f(x) = x2 + 3x + 7
f(x) = x2 + 5x + 9

Step 1: Press .

Step 2: Press the diamond key and then F1 to enter into the y=editor. 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: Press x ^ 2 + 3 x + 7.

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

Step 5: Press x ^ 2 + 5 x + 9.

Step 6: Press ENTER to enter the second function into the “y2 =” slot.

Step 7: Press HOME.

Step 8: Press the diamond key and then F3 to view the graph. You will see that the two graphs intersect. 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, arrow to the left of the intersection and 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 to access the Trace feature.

Step2: Press < or > 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 Intersection of Two Lines Algebraically

Sample problem: Find the intersection for the equations 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 – 2x = -1 – 2
x = -3
The x-intersection is -3.

Step 3: Use the value you found in Step 2 to find y.
y = 3(-3) + 2 = -7
The intersection for the two lines is (-3, -7)

Tip: If you perform the steps by hand, you can

• use an online graphing calculator like this one to check your work. It works much in the same way as the TI-89 (albeit with stripped down features). Input your two equations, and find the intersection.
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