 # 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 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.

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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