Linear algebra (impact of solution of theory of linear equations)

[hoju]

I am currently taking the first college level linear algebra course and have a question.

Consider the system of equations:

1 3 2 |1
0 -4 5 |-23
2 2 9 |t

Find values of t for which solutions to this augmented matrix can be obtained. Explain the implications of this example in the theory of linear equations.

Answer:

I found that t=-21 and that the solution is of the form {x, y, z} = {-1/4(23z+65), 1/4(5z+23), z}. That much is correct. What I don't know is what the bold statement above is asking of me. Any help is appreciated. Thanks.

[Mark44]

In row-reducing this augmented matrix, I found that if t = -21, the system has no solution. For a different value of t, I found the solution you show.

For implications of this example in the theory of linear equations, I'm not sure which implications the text author has in mind, but I would start with the geometric significance of your solution set.

[hoju]

Thanks for the reply Mark44. Here is the work I did to find t.

1 3 2 |1
0 -4 5 |-23
2 2 9 |t

row reducing

1 3 2 |1
0 -4 5 |-23
0 -4 5 |t-2

1 3 2 |1
0 -4 5 |-23
0 0 0 |t-2+23

1 3 2 |1
0 -4 5 |-23
0 0 0 |t+21

It follows that t=-21 is necessary for the system to have a solution so that 0=0 in the last row. Did I miss something?

I know that the system has an infinite number of solutions that depend on the value of z, but I am still not sure how this impacts the theory of linear equations. Probably because I am somewhat unfamiliar with the theory itself. The book isn't much help.

I am guessing that the theory of linear equations states that the roots of each equation in the system are the same if the system has a solution, but in this case the existence and identity of the solution are restricted by t and z, respectively. Would that be on the right track? Thanks for the help.