Exploring Solutions for a System of Linear Equations with Variable Coefficients

[lol_nl]

Homework Statement

How do I show that a system of linear equations either has a solution or has multiple solutions?

Homework Equations

Show that the system of equations
a*x _{1} + 2*x_{2} + a*x_{3} = 5a
x _{1} + 2*x_{2} + (2-a)*x_{3} = 5a
3*x _{1} + (a+2)*x_{2} + 6*x_{3} = 15

is solvable for every value of a. Solve the system for those values of a with more than one solution. Give a geometric interpretation of the system of equations and its solutions.

The Attempt at a Solution

I tried Gauss(-Jordan) elimination but because of the a's I could not get a nice solution.

How am I supposed to show that there is a solution regardless of what value of a I choose?

If there is more than one solution, at least one of the equations should be dependent on the other, so I should be able to reduce at least one row to all zeros? But there are multiple ways in which one or more of the equations can be a linear combination of the other two, right? E.g. I discovered that for a=1, eq. 1 and eq. 2 become equal. But how do I find all solutions?

 

[vela]

I think you're just going to have to raise your threshold of pain and row-reduce the matrix.

I'll note, however, that there isn't a solution for every a. I find for two values of a, the system is inconsistent, so there is no solution.

 

[HallsofIvy]

It's not too hard to show that there are three values of a for which the determinant of the matrix of coefficients is 0. Then put those specific values into determine whether there a none or an infinite number of solutions.