Let's solve the system of linear equations:

\[
\begin{aligned}
1. & \quad -2x + 3y + 6z = 15 \\
2. & \quad x - 2y - 5z = -28 \\
3. & \quad y + 4z = -24 
\end{aligned}
\]

### Step 1: Solve the third equation for \( y \):
The third equation is:
\[
y + 4z = -24
\]
\[
y = -24 - 4z
\]

### Step 2: Substitute \( y = -24 - 4z \) into the first and second equations.

Substitute into the first equation:
\[
-2x + 3(-24 - 4z) + 6z = 15
\]
\[
-2x - 72 - 12z + 6z = 15
\]
\[
-2x - 72 - 6z = 15
\]
\[
-2x - 6z = 87
\]
\[
x + 3z = -43.5 \quad \text{(Equation 4)}
\]

Substitute into the second equation:
\[
x - 2(-24 - 4z) - 5z = -28
\]
\[
x + 48 + 8z - 5z = -28
\]
\[
x + 3z = -76 \quad \text{(Equation 5)}
\]

### Step 3: Solve the system of equations (4) and (5):
The equations obtained are:
\[
x + 3z = -43.5
\]
\[
x + 3z = -76
\]

These two equations are contradictory (they imply two different values for the same expression). Hence, **no solution exists** for this system of equations. This system is **inconsistent**.

Would you like further details or have any questions?

Here are 8 related questions for further exploration:

1. How can we determine if a system of linear equations has no solution, one solution, or infinitely many solutions?
2. What does it mean geometrically when a system of equations has no solution?
3. How can we solve systems of equations with more than three variables?
4. What are the conditions for a system of linear equations to have a unique solution?
5. How can we interpret an inconsistent system of linear equations?
6. What is the significance of Gaussian elimination in solving linear systems?
7. How do you determine the rank of a matrix derived from a system of linear equations?
8. Can a system of nonlinear equations also be inconsistent? If so, how?

**Tip:** When solving systems of equations, checking for contradictions early can save time and clarify whether the system is consistent or inconsistent.

Learn how to solve the system of linear equations -2x + 3y + 6z = 15, x - 2y - 5z = -28, y + 4z = -24 with a detailed step-by-step solution. Understand the concept of inconsistency in linear systems and the application of Gaussian elimination. Suitable for Grades 10-12.

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