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Free_Kalibri [48]
3 years ago
7

Find the set of solutions for the linear system.

Mathematics
1 answer:
Illusion [34]3 years ago
8 0

Answer:

The system has infinitely many solutions.

\left\begin{array}{ccc}x_1&=&-\frac{1}{3}x_2-\frac{16}{9}x_4-\frac{2}{9}  \\x_2&=&s_1\\x_3&=&-\frac{4}{3}x_4+\frac{1}{3}  \\x_4&=&s_2\end{array}\right

Step-by-step explanation:

To find the solution for this system of linear equations -3x_1-x_2+4x_3=2\\-3x_3 - 4x_4 = -1you 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]

The augmented matrix can be transformed by a sequence of elementary row operations to the matrix.

There are three kinds of elementary matrix operations.

  1. Interchange two rows (or columns).
  2. Multiply each element in a row (or column) by a non-zero number.
  3. 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]

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]

which corresponds to the system

x_1+\frac{1}{3}x_2+ \frac{16}{9}x_4=-\frac{2}{9} \\x_3+ \frac{4}{3}x_4=\frac{1}{3}

We see that the variables x_2, x_4 can take arbitrary numbers; they are called free variables. Let x_2=s_1, x_4=s_2. All solutions of the system are given by

\left\begin{array}{ccc}x_1&=&-\frac{1}{3}x_2-\frac{16}{9}x_4-\frac{2}{9}  \\x_2&=&s_1\\x_3&=&-\frac{4}{3}x_4+\frac{1}{3}  \\x_4&=&s_2\end{array}\right

The system has infinitely many solutions.

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