Problem:
How many ordered pairs of real numbers satisfy the following system of equations?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
The graph of the system is shown below.
The graph of the first equation is a line with -intercept and -intercept . To draw the graph of the second equation, consider the equation quadrant by quadrant. In the first quadrant and , and thus the second equation is equivalent to , which in turn is equivalent to . Its graph consists of the rays with endpoints and , as shown. In the second quadrant and . The corresponding graph is the reflection of the first quadrant graph across the -axis. The rest of the graph can be sketched by further reflections of the first-quadrant graph across the coordinate axes, resulting in the figure shown. There are intersection points: , and , as shown.
The second equation implies that or . There are four cases:
If , then , so .
If , then , so .
If , then , so again .
If , then , so .
It may be checked that each of these ordered pairs actually satisfies the given equations, so the total number of solutions is .
The problems on this page are the property of the MAA's American Mathematics Competitions