Hello, AlgebraicallyChallenged!

I would have assumed that you've run into these cases by now.

There are three possible outcomes.Describe all possible outcomes of solving a system of two equations.

Give examples.

[1] The system has a unique solution.

Example: .

The solution is: .

The problem asks for two numbers whose sum is 8 and whose difference is 2.

. . The numbers are 5 and 3.

If we graph the two lines, they intersect at (5,3).

[2] The system has no solutions.

Example: .

If we try to solve the system, we get afalsestatement, like: .

. . This indicates that the system has no solution.

The problem asks for two numbers whose sum is 4whose sum is 3.and

. . Of course, there are no such numbers.

If we graph the two line, we find that they are parallel.

. . They donotintersect.

[3] The system has infinite soluions.

Example: .

If we try to solve the system, we get atruestatement, like: .

. . This indicates that the system has infinitely many solutions.

The problem says: find two numbers whose sum is 4

. . and if we double the numbers and add them, we get 8.

We see that is a solution . . . but so is

If we graph the lines, we get the same line twice.

. . So, of course, they "intersect" each other a zillion times.