Question

How many solutions does the following system of equations have?
2x + 5z = 10
3y + z = 2
5x - 2y = 3

Looking for a method using matrix algebra

Answer 1

First we need to create the "augmented matrix"
2 0 5 10
0 3 1 2
5 -2 0 3

I'm assuming that you want to apply Gauss-Jordan row operations to solve this system for x, y and z. Is that correct? If so, tell me a bit about your previous experience with Gauss-Jordan. This method is not terribly hard, but it does involve a lot of detail, so I'd prefer to buildon what you already know.

But wait! This is for the SOLUTION (if there is one) of the system.
Your task was to determine how many solutions the system has, not to find the actual solution.

There are 3 possible cases:
Case 1: No solution (this occurs when the last row of the matrix is a false algebraic statement, such as 0 0 0 5 (this says that 0x + 0y + 0z = 5, which is false)

Case 2: One unique solution when the last row of the matrix looks like 0 0 1 a, in which case the z coordinate of the solution would be a.

Case 3: Infinite number of solutions when the last row of the matrix is all zero: 0 0 0 0.

for more detail, please do an internet search for "matrix solutions of systems of linear equations."

Answer 2

Please see https://people.richland.edu/james/lecture/m116/matrices/matrices.html
This provides considerable information on matrix solutions, but doesn't say a lot about determining how many solutions your system will have. But look at "inconsistent" near the bottom.

Answer 3

Thank you so much for your response - I really appreciate it!

Answer 4

DC: My great pleasure!