Graphing calculators are one way to help students understand the relationship between graphs and the solution of a set of equations. The key to understanding that relationship is knowing that the equations' solution is the intersection point of the graphs of the individual equations. Finding the intersection point of two equations requires a graphing calculator that allows you to enter two or more equations. After you enter and graph the equations, you must then look for the point or points where the two graphs intersect. That point or points, expressed in x and y coordinates, will be the equations' solution.

Use the equation of a parabola (a U shaped graph) for the first equation. For this example, use the parabola equation y=x^2. Type the right side of the equation, x^2, into the first function(equation) text box on your calculator.

Use the equation of a line for the second equation. For this example, use the equation y=x. Type the right side of the equation, x, into the second function(equation) text box on your calculator.

## Sciencing Video Vault

Select the "graph" or "plot" function of your calculator. Observe that two graphs, one of the parabola and one of the line, are graphed on the display. Note that the line and the parabola intersect at the points (0,0) and (1,1). Write down that the solution set of the two equations, y=x^2 and y=x, is defined by the points (0,0) and (1,1).

Substitute x=0 into both equations, y=x^2 and y=x, to verify that the value of y for x=0 is 0 for both equations. Substitute the x=1 into the two equations to verify that the value of y for x=1 is 1 for both equations. Conclude that the solution is correct because the two values of x( 0 and 1) produce the same value of y(0 and 1) in the two equations.

#### Tip

Use the 2D calculator from FooPlot listed in the resources section if you do not have a calculator of your own. Select the "Intersection" button, and then click the intersection point to display the exact value of the x and y coordinates of the solution. Save the file with the save buttons.

#### Warning

If you don't see the intersection point of the graphs, try panning across the display or reset the scales of your graph so that you can see more of the graph. Desktop calculators, because of their small screens, often require that you approximate the solution first so that you can set a window that covers the region where the graphs intersect.