Question

For each of the following systems of linear equations, use Gauss-Jordan elim-

ination to find the set of solutions. Write the set of solutions in vector form.

Assume in each case that the set of variables has the form x 1 , . . . , x n (so, for instance, x 1 is a variable even if it does appear in any of the equations).

4. x2 + 2x3 -x5 = 6

Answer 1

The only equation given is x₂+2x₃-x₅ = 6.

Then, we have x₂ = 6-2x₃+x₅ so that (x₁,x₂,x₃,x₄,x₅) = (x₁, 6-2x₃+x₅ ,x₃,x₄,x₅) = (0,6,0,0,0)+ x₁(1,0,0,0,0) + x₃(0,-2,1,0,0)+x₄(0,0,0,1,0) +x₅(0,1,0,0,1). Now, let x₁ = r, x₃ = s, x₄ = t and x₅ = u. Then, the vector form of the general solution is (x₁,x₂,x₃,x₄,x₅) = (0,6,0,0,0)+ r(1,0,0,0,0) + s(0,-2,1,0,0) +t(0,0,0,1,0) +u(0,1,0,0,1) where r, s, t, u are arbitrary real numbers.

Note:

The Gauss-Jordan elimination can be used where there are more than 1 linear equations.