Solving a system on linear equations

[noamo48]

Homework Statement

Row A) x₁ + 2x₂ - x₃ + x₄ + 2x₅ = 1
Row B) 2x₁ + x₂ + x₃ - x₄ + 2x₆ = -1
Row C) x₁ + 4x₂ - 2x₃ + x₄ - x₅ = 0
Row D) x₁ + x₂ + 3x₃ + x₄ + x₆ = 2

Homework Equations

R_i <--> R_j
R_i --> cR_j, c not equal to 0
R_i --> R_i + cR_j, c \neq 0, i \neq j

The Attempt at a Solution

1) Swap rows B and C

2a) Add (-1) x Row A to Row B
2b) Add (-2) x Row A to Row C
2c) Add (-1) x Row A to Row D

3a) Multiply Row B by (1/2)
3b) Add (-2) x Row B to Row A
3c) Add (3) x Row B to Row C
3d) Add Row B to Row D

4a) Multiply Row C by (2/3)
4b) Add (1/2) x Row C to Row B
4c) Add (-7/2) x Row C to Row D

5a) Multiply Row D by (1/7)
5b) Add (-1) x Row D to Row A
5c) Add Row D to Row B
5d) Add (2) x Row D to Row C

I wind up with a very funky solution set full of large fractions as coefficients...can someone let me know what they get please so I can compare...thanks!

 

[hotvette]

The first thing you need to recognize is that there are more variables (x₁-x₆) than there are equations. This means there are either no solutions or an infinite number of solutions. Are you sure the problem statement is correct?

 

[noamo48]

I am positive that is the problem in my book...

 

[hotvette]

I don't know how your instructor would want you to approach this problem but I'd solve it by moving x₅ and x₆ to the right hand side and solve the resulting 4 x 4 problem recognizing that x₅ and x₆ can take on arbitrary values (meaning there are infinitely many solutions).