Question

The following system of equation has : x-y-z=4,2x-2y-2z=8, 5x-5y-5z=20​

Answer 1

Answer:

x – y – z = 4

2x – 2y – 2z = 8

5x – 5y – 5z = 20

Answer 2

Answer:

The system of equations $x-y-z=4,2 x-2 y-2 z=8,5 x-5 y-5 z=20$ has infinitely many solutions.

Step-by-step explanation:

We have the system of equations:

$x-y-z=4 \\ 2 x-2 y-2 z=8 \\ 5 x-5 y-5 z=20$

Solve the system of equations using Gaussian Elimination.

Write the system of equation in matrix form as shown below:

$\left[\begin{array}{ccc}1 & -1 & -1 \\ 2 & -2 & -2 \\ 5 & -5 & -5\end{array}\right]\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{c}4 \\ 8 \\ 20\end{array}\right]$

Multiply Row 1 by $-2$:

$\left[\begin{array}{ccc}-2 & 2 & 2 \\ 2 & -2 & -2 \\ 5 & -5 & -5\end{array}\right]\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{c}-8 \\ 8 \\ 20\end{array}\right]$

Then add Row 1 and Row 2:

$\left[\begin{array}{ccc}0 & 0 & 0 \\ 2 & -2 & -2 \\ 5 & -5 & -5\end{array}\right]\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{c}0 \\ 8 \\ 20\end{array}\right]$

The 1st row of the matrix gets reduced $0$.

Hence, there are infinitely many solutions.