Graphing

A system of equations is two or more linear equations that model the same situation. The solution to a linear system of equations is the point which lies on both lines. In other words, the solution is the point where the two lines intersect.

One way to solve a system of equations is by graphing the lines on the same coordinate plane and reading the intersection point from the graph. This method most often offers approximate solutions. Exact answers are found in the rare case when the x and yvalues of the solution are integers. However, this method is great at offering a visual representation of the system of equations and demonstrates that the solution to a system of equations is the intersection of the two lines.

Solve the following system of equations by graphing.

Since these two lines are in standard form, finding intercepts is an appropriate method of graphing.

Line 1

2x +3y = 6

When x = 0, y = 2 which results in (0,2)

When y = 0, x = 3 which results in (3, 0)

Line 2

4x -y = -2

When x = 0, y = 2 which results in (0, 2)

When y = 0, x = which results in ( , 0)

The graph shows that the lines intersect at (0,2).

Therefore the solution is x = 0, y = 2.

A graphing calculator can be used to find or check solutions to a system of equations. It can be used as an alternative to graphing the equations by hand. The following example will show graphing using a TI-84 calculator.

Example Solve the following system of equations using a graphing calculator.

Line 1:

Line 2:

Option 1: Use [TRACE] and move the cursor with the arrows until it is on top of the intersection point. The values of the coordinate point will be on the bottom of the screen. Using the window above, the values will be x = 4.0957447 and y = 0.03191489. These are approximations due to the pixels on the graph screen.

Option 2: Look at the table of values by pressing [2nd][GRAPH]. The screen below shows a table of values for this system of equations. Scroll down until the values for y₁ and y₂ are the same. In this case, this occurs at x = 4 and y = 0.

Option 3: Using the [2nd][TRACE] function will allow you to calculate a value.

• Scroll down and select intersect.
• The calculator will display the graph with the question [FIRSTCURVE]? Move the cursor along the first curve until it is close to the intersection and press [ENTER].
• The calculator now displays [SECONDCURVE]?
• Move the cursor to the second line and press [ENTER].
• The calculator now displays [GUESS]?
• Press [ENTER] and the calculator displays the solution at the bottom of the screen

The point of intersection is x = 4 and y = 0.