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The given system of linear equations is:

\[
\begin{cases}
2x - y + 4z = 33 \\
x + 2y - 3z = -26 \\
-5x - 3y + 5z = 23
\end{cases}
\]

To solve this system, we can use the method of elimination or substitution. Let's use elimination here.

**Step 1: Eliminate \(x\) from the second and third equations**

From the first equation, multiply by 5 to match the \(x\) coefficient in the third equation:
\[ 10x - 5y + 20z = 165 \]

Now we add this to the third equation:
\[ 10x - 5y + 20z + (-5x - 3y + 5z) = 165 + 23 \]
\[ 5x - 8y + 25z = 188 \quad \text{(Equation 4)} \]

**Step 2: Eliminate \(x\) from the first and second equations**

Multiply the first equation by 1 to keep it unchanged:
\[ 2x - y + 4z = 33 \]

Now multiply the second equation by 2 to match the \(x\) coefficient:
\[ 2x + 4y - 6z = -52 \]

Subtract the first equation from this result:
\[ (2x + 4y - 6z) - (2x - y + 4z) = -52 - 33 \]
\[ 5y - 10z = -85 \quad \text{(Equation 5)} \]

**Step 3: Solve the system of the resulting equations (Equations 4 and 5)**

Equation 4: \( 5x - 8y + 25z = 188 \)

Equation 5: \( 5y - 10z = -85 \)

From Equation 5:
\[ y - 2z = -17 \]
\[ y = 2z - 17 \]

**Step 4: Substitute \( y = 2z - 17 \) into Equation 4**

\[ 5x - 8(2z - 17) + 25z = 188 \]
\[ 5x - 16z + 136 + 25z = 188 \]
\[ 5x + 9z = 52 \]
\[ x = \frac{52 - 9z}{5} \]

**Step 5: Back-substitute \( x \) and \( y \) into the original equations**

Now substitute \( x = \frac{52 - 9z}{5} \) and \( y = 2z - 17 \) back into one of the original equations to solve for \( z \). We use the first equation:
\[ 2\left(\frac{52 - 9z}{5}\right) - (2z - 17) + 4z = 33 \]
\[ \frac{104 - 18z}{5} - 2z + 17 + 4z = 33 \]
\[ \frac{104 - 18z + 85 + 20z}{5} = 33 \]
\[ \frac{189 + 2z}{5} = 33 \]
\[ 189 + 2z = 165 \]
\[ 2z = -24 \]
\[ z = -12 \]

Now, find \( x \) and \( y \) using \( z = -12 \):
\[ y = 2(-12) - 17 = -24 - 17 = -41 \]
\[ x = \frac{52 - 9(-12)}{5} = \frac{52 + 108}{5} = \frac{160}{5} = 32 \]

**Solution:**
\[ x = 32, \quad y = -41, \quad z = -12 \]

Would you like further details or have any questions?

Here are some related questions you might find interesting:
1. How can we verify the solution of a system of linear equations?
2. What is the geometric interpretation of a system of three linear equations in three variables?
3. How does Gaussian elimination work for solving systems of equations?
4. Can we use matrix methods such as determinants or inverses to solve this system? How?
5. What are some common applications of solving systems of linear equations?

**Tip:** When solving systems of equations, always check your solutions by substituting them back into the original equations to ensure they satisfy all the equations.

Solve the system of equations: 2x - y + 4z = 33, x + 2y - 3z = -26, -5x - 3y + 5z = 23

Learn how to solve the system of linear equations: 2x - y + 4z = 33, x + 2y - 3z = -26, -5x - 3y + 5z = 23 using the elimination method. This step-by-step solution is suitable for students in Grades 9-12.

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