Question

Solve the system using matrices. Gaussian elimination with back-substitution or Gauss-Jordan elimination.

X + 5y= 0
X + 6y + z = 1
2x - y - z= -21

The solution set is _,_,_ si

Answer 1

In augmented matrix form, the system is

`\begin{bmatrix}1&5&0&0\\1&6&1&1\\2&-1&-1&-21\end{bmatrix}`

Subtract row 1 from row 2, and subtract 2 times row 1 from row 3:

`\begin{bmatrix}1&5&0&0\\0&1&1&1\\0&-11&-1&-21\end{bmatrix}`

Add 11 times row 1 to row 3:

`\begin{bmatrix}1&5&0&0\\0&1&1&1\\0&0&10&-10\end{bmatrix}`

Divide through row 3 by 10:

`\begin{bmatrix}1&5&0&0\\0&1&1&1\\0&0&1&-1\end{bmatrix}`

Row 3 says
`z=-1`. Substituting into row 2, we get
`y+z=1\implies y=2`. Substituting both into row 1, we get
`x+5y=0\implies x=-10`. So the solution is the single ordered triplet (-10, 2, -1).