Question

consider this augmented matrix, which represents a system of linear equations.

\\\

$$\begin{bmatrix} 1 & 0 & 2 & 0 & -3 \\\\ 0 & 1 & 1 & 0 & -2 \\\\ 0 & 0 & 0 & 1 & 5 \\end{bmatrix}$$

\\

write the general solution for this system:

variable \\(x_1\\)
\\(x_1 = \\)
\\(x_1\\) is free

variable \\(x_2\\)
\\(x_2 = \\)
\\(x_2\\) is free

variable \\(x_3\\)
\\(x_3 = \\)
\\(x_3\\) is free

variable \\(x_4\\)
\\(x_4 = \\)
\\(x_4\\) is free

Explanation:

Identify the system of equations

Using the Reduced Echelon Form and System Solution Set knowledge points

$$
LATEXBLOCK0
$$

Determine the free variables

Using the Echelon Form knowledge point

• Pivots are in columns 1, 2, and 4.
• Column 3 has no pivot, so \(x_3\) is free.

Express basic variables in terms of free variables

Using the System Solution Set knowledge point

$$
LATEXBLOCK1
$$

Answer:

Consider this augmented matrix, which represents a system of linear equations.

$$
LATEXBLOCK0
$$

Write the general solution for this system:

Variable \(x_1\)

• \(x_1 =\) <blank>\(-3 - 2x_3\)</blank> (Correct answer)
• \(x_1\) is free

Variable \(x_2\)

• \(x_2 =\) <blank>\(-2 - x_3\)</blank> (Correct answer)
• \(x_2\) is free

Variable \(x_3\)

• \(x_3 =\) <blank></blank>
• \(x_3\) is free (Correct answer)

Variable \(x_4\)

• \(x_4 =\) <blank>\(5\)</blank> (Correct answer)
• \(x_4\) is free