Systems of Equations & Solutions

Question

For each of the following systems of equations, state if the system has a unique solution, an infinite number of solutions, or is inconsistent (no solutions). If the system has a unique solution, exhibit the solution. If the system has an infinite number of solutions, exhibit three different solutions.

x + y + z + w = 0

4x + 3y + 5z - w = -5

2x + 5y + 3z + 9w = 3

3x - 4y - 2z + 3w = 1

5x + 2y + 3z - 4w = 1

Answer 1

I used the TI calculator RREF and got the following:

1 0 0 0 0

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

0 0 0 0 1

Looking at the last line, that tells me 0z=1, which is not possible. So the answer would be no solution.

Answer 2

If you calculate the determinant for the first four equations, we see that it is not 0, so it will have a unique solution.

The solution is x = 1; y = 2; z = -3; w = 0

When we plug this solution into the fifth equation, it does not work. We get 0, not 1; thus, there is no solution.