Systems of Equations Solved Graphically

Graphs can be used to solve systems of equations. This method, however, usually allows only approximate solutions, whereas the algebraic method arrives at exact solutions.

Example 1

Solve the following system of equations graphically.

  • (1)

    x 2 + 2 y 2 = 10

  • (2)

    3 x 2y 2 = 9

Equation (1) is the equation of an ellipse. Convert the equation into standard form.

equation

The major intercepts are at equation and equation, and the minor intercepts are at equation and equation.

Equation (2) is the equation of a hyperbola. Convert the equation into standard form.

equation

The transverse axis is horizontal, and the vertices are at equation and equation, as shown in Figure 1.

The approximate answers are equation

The exact answers are equation

Refer to Example for the algebraic approach to this problem; it gives the exact answers.

Figure 1. Approximate solutions to hyperbola and ellipse.

figure

Example 2

Solve the following system of equations graphically.

  • (1)

    x 2 + y 2 = 100

  • (2)

    xy = 2

Equation (1) is the equation of a circle centered at (0, 0) with a radius of 10. Equation (2) is the equation of a line. The solutions are

{(–6, –8), (8, 6)}

The graph is shown in Figure 2.

Refer to Example for the algebraic approach to this problem.

Figure 2. Circle with intersecting line.

figure