Solving Systems of Equations by Graphing Calculator

Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

Plotting the curves for a system of equations lets us see where the curves cross. These intersections are the solutions. In this lesson, we use a graphing calculator to enter functions, plot curves and find intersections.

Looking for intersections and solutions to equations
two country paths crossing

It's time to head out and meet your friend. Each of you is walking along a separate path. You keep walking until you reach the crossing of the two paths. This intersection is part of each path. It's like the intersection of two curves. It's the one place belonging to both curves. When solving a system of equations, the intersection is the solution. In this lesson, we use a graphing calculator to display the plots of curves, find their intersection and get a solution.

Getting Started

To get started, let's bring up the function screen. On my graphing calculator, this is labeled as Y=. This is an alternative command located above the F1 button. The words are written in green. To access this command, press the Green Diamond and then F1. There should be an equivalent function on your graphing calculator.

Access the function screen
the Green Diamond and then F1

You should see this.

This is the place for entering functions
showing the function screen

To see how everything works, let's solve these two equations.

the two equations to be solved

We are looking for the value of x and the value of y that make both of these equations true. As a plot, the coordinates at the intersection of the two curves is the solution. Type in the equations and press ENTER after each equation.

This is the function screen after entering the two equations
after entering the two equations

To plot these two functions, press F2 and select 4. The F2 button performs a zoom plot and the 4 is 'ZoomDec,' which stands for zoom decimal. 'Decimal' means the x-axis and the y-axis are subdivided into 0.1 increments.

This is the zoomed plot of the two functions

To find the intersection, press F5 and select 5. Then, press ENTER twice. We now can enter the Lower Bound. This means a value for x to the left of the intersection point. We type .1 and press ENTER.

Set the Lower Bound
setting the Lower Bound

Then, we need an Upper Bound. This is a value for x to the right of the intersection point. We type 3 and press ENTER.

Set the Upper Bound
setting the Upper Bound

After pressing ENTER, we get this graph.

The plot of the intersection of the two functions

The intersection of the two curves is at x = 2 and y = 1. This is our solution.

Selecting and De-selecting

Let's add another equation to our list: y = x^2 - 3.

Add a third equation to the list
adding another equation

As before, we can plot by pressing F2, followed by selecting 4.

Zoom plot the 3 equations
zoom plotting the 3 equations

But this time, we see three functions plotted at the same time. There is one solution where all three curves cross. But, there are other solutions as well. To makes things clearer, we can select which functions to be plotted.

Returning to the function screen by pressing Green Diamond and F1, we see a check-mark just to the left of each function. A check-mark means the function is selected. To select or deselect, we move the cursor to the desired equation and press the F4 button. The up and down arrow keys move the cursor. In this case, let's deselect the first equation. Here's the screen after pressing the F4 button. Notice the check-mark is absent.

Deselect the first equation
deselecting the first equation

Now, when we press F2 and 4 we see only the curves y = 3 - x and y = x^2 - 3.

Zoom plot the selected curves
zoom plotting the selected curves

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