Answer:
The system has infinitely many solutions.

Step-by-step explanation:
To find the solution for this system of linear equations
you must:
Step 1: Transform the augmented matrix to the reduced row echelon form.
A matrix is a rectangular arrangement of numbers into rows and columns.
A system of equations can be represented by an augmented matrix.
In an augmented matrix, each row represents one equation in the system and each column represents a variable or the constant terms.
This is matrix that represents the system
![\left[ \begin{array}{ccccc} -3 & -1 & 4 & 0 & 2 \\\\ 0 & 0 & -3 & 4 & -1 \end{array} \right]](https://tex.z-dn.net/?f=%5Cleft%5B%20%5Cbegin%7Barray%7D%7Bccccc%7D%20-3%20%26%20-1%20%26%204%20%26%200%20%26%202%20%5C%5C%5C%5C%200%20%26%200%20%26%20-3%20%26%204%20%26%20-1%20%5Cend%7Barray%7D%20%5Cright%5D)
The augmented matrix can be transformed by a sequence of elementary row operations to the matrix.
There are three kinds of elementary matrix operations.
- Interchange two rows (or columns).
- Multiply each element in a row (or column) by a non-zero number.
- Multiply a row (or column) by a non-zero number and add the result to another row (or column).
Using elementary matrix operations, we get that
Row Operation 1: Multiply the 1st row by -1/3
Row Operation 2: Multiply the 2nd row by -1/3
Row Operation 3: Add 4/3 times the 2nd row to the 1st row
![\left[ \begin{array}{ccccc} 1 & \frac{1}{3} & 0 & \frac{16}{9} & - \frac{2}{9} \\\\ 0 & 0 & 1 & \frac{4}{3} & \frac{1}{3} \end{array} \right]](https://tex.z-dn.net/?f=%5Cleft%5B%20%5Cbegin%7Barray%7D%7Bccccc%7D%201%20%26%20%5Cfrac%7B1%7D%7B3%7D%20%26%200%20%26%20%5Cfrac%7B16%7D%7B9%7D%20%26%20-%20%5Cfrac%7B2%7D%7B9%7D%20%5C%5C%5C%5C%200%20%26%200%20%26%201%20%26%20%5Cfrac%7B4%7D%7B3%7D%20%26%20%5Cfrac%7B1%7D%7B3%7D%20%5Cend%7Barray%7D%20%5Cright%5D)
Step 2: Interpret the reduced row echelon form
The reduced row echelon form of the augmented matrix is
![\left[ \begin{array}{ccccc} 1 & \frac{1}{3} & 0 & \frac{16}{9} & - \frac{2}{9} \\\\ 0 & 0 & 1 & \frac{4}{3} & \frac{1}{3} \end{array} \right]](https://tex.z-dn.net/?f=%5Cleft%5B%20%5Cbegin%7Barray%7D%7Bccccc%7D%201%20%26%20%5Cfrac%7B1%7D%7B3%7D%20%26%200%20%26%20%5Cfrac%7B16%7D%7B9%7D%20%26%20-%20%5Cfrac%7B2%7D%7B9%7D%20%5C%5C%5C%5C%200%20%26%200%20%26%201%20%26%20%5Cfrac%7B4%7D%7B3%7D%20%26%20%5Cfrac%7B1%7D%7B3%7D%20%5Cend%7Barray%7D%20%5Cright%5D)
which corresponds to the system

We see that the variables
can take arbitrary numbers; they are called free variables. Let
,
. All solutions of the system are given by

The system has infinitely many solutions.