3 Questions Regarding a particular system of unknowns x,y,z

[bmed90]

Homework Statement

Ok the system we are working with is

x-2y =1
x-y+az=2
ay+9z=b

Homework Equations

1.)> For which values of a does the system have a unique solution?

2.)> For which pairs of values (a,b) does the system have more than one solution?

3.)>Why is it that the value of b has no effect on whether or not the system has a unique solution?

The Attempt at a Solution

Im going to be honest and say I don't really know where to begin on any of these. Is this a system of 3 equations with 5 unknowns? I know that's kind of a dumb question but I'm just making sure a and b are two variables aside from x y and z. Any help is apreciated

 

[Mark44]

Homework Statement

Ok the system we are working with is

x-2y =1
x-y+az=2
ay+9z=b

Homework Equations

1.)> For which values of a does the system have a unique solution?

2.)> For which pairs of values (a,b) does the system have more than one solution?

3.)>Why is it that the value of b has no effect on whether or not the system has a unique solution?

The Attempt at a Solution

Im going to be honest and say I don't really know where to begin on any of these. Is this a system of 3 equations with 5 unknowns? I know that's kind of a dumb question but I'm just making sure a and b are two variables aside from x y and z. Any help is apreciated

You should consider this to be a system of 3 equations in 3 variables. a and b are parameters - fixed numbers that don't happen to be known.

 

[bmed90]

Ok so should I solve for x, y, and x in terms of A and B then Solve for A and B ?

 

[Mark44]

Not quite. First, solve for x, y, and z. You are NOT solving for a and b, just determining what values of a and b let you get a unique solution or multiple solutions.

 

[bmed90]

Ok so I am going to try to solve for x y z using the Gaussian method. This will give me x y z in terms of A and B. Will this be my final answer? (will this be determining what values of a and b let you get a unique solution or multiple solutions)

 

[vela]

Not really. When you row reduce a matrix, how do you tell when you have a unique solution, an infinite number of solutions, and no solution? What you want to do is row-reduce your matrix and figure out which values of a and b will yield those conditions.

 

[HallsofIvy]

You will, eventually, have to divide by something involving a. There will be a unique solution as long as that is not 0. There will be more than one solution if both the number you are diving by and the number you are dividing are 0.