Linear Algebra - Find an equatio relating a,b,c

[jinksys]

Homework Statement

http://i.imgur.com/FuMGN

Homework Equations

The Attempt at a Solution

I know that for a linear system to be consistent it must have one or more answers.

After some row operations I get this matrix: http://i.imgur.com/KTEkF

I know that if I have a row of zeros that I have a trivial solution, so do I need to equate b-a-c to zero?

B - A - C = 0?

 

[Mark44]

Think about what the augmented matrix represents: Ax = c, where A is your 3 x 3 matrix, x is the vector <x, y, z>, and c is the vector of constants, <a, b, c>.

The bottom row means 0x + 0y + 0z = b - a - c. Regardless of the values of x, y, and z, what is the only value that the right side could have to make an equation that had a solution?

 

[jinksys]

Think about what the augmented matrix represents: Ax = c, where A is your 3 x 3 matrix, x is the vector <x, y, z>, and c is the vector of constants, <a, b, c>.

The bottom row means 0x + 0y + 0z = b - a - c. Regardless of the values of x, y, and z, what is the only value that the right side could have to make an equation that had a solution?

Zero. So the relationship between a,b, and c is b-c-a=0, correct?

 

[Mark44]

Right. So the system of equations is consistent provided that b - c - a = 0. If two variables are specified, the third can be found. That means that there are two free variables, and therefore, the set of points in R³ for which the system of equations is consistent is a plane in space.

 

[jinksys]

Right. So the system of equations is consistent provided that b - c - a = 0. If two variables are specified, the third can be found. That means that there are two free variables, and therefore, the set of points in R³ for which the system of equations is consistent is a plane in space.

Thank you for the help.