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 2 – y 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)
x – y = 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.
