# System of linear equations

## 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?

## Answers and Replies

vela
Staff Emeritus
Science Advisor
Homework Helper
Education Advisor
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
Science Advisor
Homework Helper
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 in to determine whether there a none or an infinite number of solutions.