Systems of Linear Equations using row echelon form. Need help with Part 2.

Question

Part 1: Verify that the following augmented matrices are in rref(row echelon form), identify the leading and the free

variables, then find all the solutions, if there are any, to the systems of equations

corresponding to the rref(row echelon form). Assume that the variables are x₁, x₂, . . . , x_n. Use the letters r,

s, t, and u for values of the free variables. Write your solutions in vector form.

[ 1 0 0 7 5 l -5 ]

0 1 0 -4 -3 l 2

0 0 1 6 -2 l 4

I got <x_1,x_2, x_3, x_4, x₅> = <-5 -7r -5s, 2+ 4r + 3s, 4 - 6r +2s, r, s > as the answer.

Part 2: Give the simplest solution for the system (i.e. where all the free variables are set to zero), and find the

particular solution which fits the given description/s, if it is possible to do so.

x₁ = −3 and x₃ = 12.

Answer 1

I believe that the answer to Part 1 is correct.

For Part 2, setting x₄ = r = 0 and x₅ = s = 0, we get x₁ = -5 - 7(0) - 5(0) = -5, x₂ = 2 + 4(0) + 3(0) = 2, and

x₃ = 4 - 6(0) + 2(0) = 4.

So, a solution is < -5, 2, 4, 0, 0 >