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.

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

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

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

The approximate answers are

The exact answers are

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

Figure 1. Approximate solutions to hyperbola and ellipse.

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.

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