# Solving Equations Graphically

## Introduction

Graphical calculators which were already introduced in another lesson allow you to solve several different types of equations graphically. These
include:
•
Linear Equations
•
Quadratic Equations
•
Inequalities
Please note that this tutorial is demonstrated on a Texas Instruments TI-84 calculator. However, this lesson also applies to most other graphic
calculators but the layout may be slightly different.
## Linear Equations

Solving linear equations on a graphic calculator is very easy. However, unless you are specifically asked to do this or the equation is very complicated
it is quicker to do it using algebraic methods.
### Solving Linear Equations Graphically

3x+1=6x-8
1.
Write the equation as
Y=3x+1 and Y=6x-8
2.
Hit Y= on your graphic calculator and type in each equation.
3.
Hit GRAPH and your calculator will sketch the graph.
4.
Hit 2ND > TRACE (CALC) and select 5: intersect
5.
You can now select the first line near the intersection. Then
press ENTER. Do the same for the second line.
6.
After you have pressed ENTER the second time, the cursor
should fix on the intersection and at the bottom of the screen
you will find the x and the y value.
7.
The x value corresponds to the value of x in the equation. You
can check this by working out the linear equation algebraically.
In this example, it is 3.
## Quadratic Equations

Quadratic equations can be solved very easily with the graphic calculator. Especially for the ones which do not factorise this is really useful. There are
several steps involved.
### Solving Quadratic Equations Graphically

x²-x-6=0
1.
Make sure the equation is arranged so that it is equal to 0.
2.
Replace 0 with Y. Y=x²-x-6
3.
Hit Y= on your graphic calculator and type in the equation.
4.
Hit GRAPH and your calculator will sketch the graph.
5.
Press 2ND > TRACE (CALC) and select 2: zero.
6.
On the graph you must select the left and right bound through
the point where the graph cuts the x axis (0). Move around the
graph using the arrow buttons and press ENTER to select the
bound.
7.
After that Guess? will appear below the graph. Press ENTER
again and you will see the x and y value of the 0. The x value
corresponds to an answer.
8.
For quadratics with two answers redo steps 5-7 for the other
intersection with the x-axis.
## Inequalities

Inequalities can be very difficult to solve especially when you meet more complex fractional inequalities. Luckily we can solve them graphically in just a
few seconds.
### Solving Inequalities Graphically

x²-x-6 < x+3
1.
Make each side of the inequality equal to Y.
Y=x²-x-6
Y=x+3
2.
Hit Y= on your graphic calculator and type in each equation.
3.
Hit GRAPH and your calculator will sketch the graph.
4.
Press 2ND > TRACE (CALC) and select 5: intersect.
5.
Use the arrow keys and ENTER to select points on the
different lines close to the intersection. When you see Guess?
press ENTER again. The x and y values will be shown
underneath the graph
6.
Do this to find both intersections (x values)
7.
Look at the inequality and the graph. In this case we are trying
to find values of x for which x+3 is greater than x²-x-6. So we
look where the red lin is above the blue line. This is the case
in-between the two blue lines.
8.
Therefore we can conclude that x²-x-6 < x+3 for -2.16 < x < 4.16 because in-between these two value the red line is above the blue line. You will
have to assess this specifically for the problem you are dealing with.
## Conclusion

These steps will allow you to solve most types of equations graphically. Make sure to check out our other lessons in the library and check
back soon for more graphic calculator tutorials.

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Chapter 36.1: Learning Outcomes
Students will learn how to solve linear functions using their graphic
calculator!
Students will learn how to solve quadratic equations using their graphic
calculator!
Students will learn how to solve inequalities using their graphic
calculator!

## VIDEO LESSON:

## NOT CURRENTLY AVAILABLE!