Question

question 33 points 2

solve:
\\
\

$$\begin{aligned} 2x - y + z &= 7 \\\\ 3x + y - 5z &= 13 \\\\ x + y + z &= 5 \\end{aligned}$$

\\

\\(x = 1, y = 1, z = 0\\)

\\(x = 4, y = 1, z = 0\\)

Explanation:

Set up the system of equations

Using the Coefficient Matrix knowledge point

$$
LATEXBLOCK0
$$

Eliminate variables to solve the system

We can eliminate \(y\) by adding Equation 1 and Equation 3:

$$
LATEXBLOCK1
$$

Next, we eliminate \(y\) by subtracting Equation 3 from Equation 2:

$$
LATEXBLOCK2
$$

Solve the simplified system for x and z

From Equation 5, we express \(x\) in terms of \(z\):

$$ x = 3z + 4 $$

Substitute this expression for \(x\) into Equation 4:

$$
LATEXBLOCK3
$$

Substitute \(z = 0\) back to find \(x\):

$$ x = 3(0) + 4 = 4 $$

Find the value of y

Substitute \(x = 4\) and \(z = 0\) into Equation 3:

$$
LATEXBLOCK4
$$

Verify the solution and match options

Using the Matrix Determinant knowledge point

$$
LATEXBLOCK5
$$

Answer:

• \(x = 1, y = 1, z = 0\)
• \(x = 4, y = 1, z = 0\) (Correct answer)