Respuesta :
Answer:
x =-2t , y = -t, z= t
Step-by-step explanation:
x+2y+z+3w=0
x-y+w=0
y-z+2w=0
The augmented system would be given by:
[tex]\begin{matrix}1 & 2 & 1 & 3 & 0\\1 & -1 & 0 & 1 & 0\\0 & 1 & -1 & 2 & 0\\\end{matrix} [/tex]
Now we can do operations in order to reduce it to the row echelon form
1) R1 *(-1) + R2
[tex]\begin{pmatrix}1 & 2 & 1 & 3 & 0\\0 & -3 & -1 & -2 & 0\\0 & 1 & -1 & 2 & 0\\\end{pmatrix} [/tex]
2) R2 *(-1/3)
[tex]\begin{pmatrix}1 & 2 & 1 & 3 & 0\\0 & 1 & 1/3 & 2/3 & 0\\0 & 1 & -1 & 2 & 0\\\end{pmatrix} [/tex]
3) R2*(-1)+R3
[tex]\begin{pmatrix}1 & 2 & 1 & 3 & 0\\0 & 1 & 1/3 & 2/3 & 0\\0 & 0 & -4/3 & 4/3 & 0\\\end{pmatrix} [/tex]
4) R3*(-3/4)
[tex]\begin{pmatrix}1 & 2 & 1 & 3 & 0\\0 & 1 & 1/3 & 2/3 & 0\\0 & 0 & 1 & -1 & 0\\\end{pmatrix} [/tex]
5) R3(-1/3) + R2; R3(-1) +R1
[tex]\begin{pmatrix}1 & 2 & 0 & 4 & 0\\0 & 1 & 0 & 1 & 0\\0 & 0 & 1 & -1 & 0\\\end{pmatrix} [/tex]
6) R2(-2) +R1
[tex]\begin{pmatrix}1 & 0 & 0 & 2 & 0\\0 & 1 & 0 & 1 & 0\\0 & 0 & 1 & -1 & 0\\\end{pmatrix} [/tex]
Let w=t a free variable then the solution is given by:
x =-2t , y = -t, z= t
