2𝑥+4𝑦−3𝑧=0 , 22𝑥+4𝑦−3𝑧=0 , 2𝑥−𝑦+𝑧=3 , 3𝑥−2𝑦+2𝑧=5

Let's analyze the system of linear equations you provided:

1. \( 2x + 4y - 3z = 0 \)
2. \( 22x + 4y - 3z = 0 \)
3. \( 2x - y + z = 3 \)
4. \( 3x - 2y + 2z = 5 \)

### Step 1: Eliminate variables by subtracting equations
First, let's subtract equation (1) from equation (2) to simplify:
\[
(22x + 4y - 3z) - (2x + 4y - 3z) = 0
\]
\[
20x = 0 \quad \Rightarrow \quad x = 0
\]

### Step 2: Substitute \( x = 0 \) into the remaining equations
Substitute \( x = 0 \) into equations (1), (3), and (4).

1. Substituting into equation (1):
\[
2(0) + 4y - 3z = 0 \quad \Rightarrow \quad 4y - 3z = 0 \quad \Rightarrow \quad y = \frac{3z}{4}
\]
   
2. Substituting into equation (3):
\[
2(0) - y + z = 3 \quad \Rightarrow \quad -y + z = 3
\]
Substitute \( y = \frac{3z}{4} \) into this equation:
\[
-\frac{3z}{4} + z = 3
\]
\[
\frac{z}{4} = 3 \quad \Rightarrow \quad z = 12
\]

### Step 3: Solve for \( y \)
Now that we have \( z = 12 \), substitute into \( y = \frac{3z}{4} \):
\[
y = \frac{3(12)}{4} = 9
\]

### Step 4: Verify with equation (4)
Substitute \( x = 0 \), \( y = 9 \), and \( z = 12 \) into equation (4):
\[
3(0) - 2(9) + 2(12) = 5
\]
\[
-18 + 24 = 5 \quad \Rightarrow \quad 6 = 5
\]
This shows there is no solution that satisfies all four equations, meaning the system is **inconsistent**.

### Conclusion:
The system of equations has **no solution**.

Would you like more details or have any questions?

Here are some related questions to explore:

1. How can we check if a system of equations is consistent or inconsistent?
2. What are the methods for solving systems of linear equations (e.g., substitution, elimination)?
3. What does it mean if a system has infinitely many solutions?
4. How does row-reduction (Gaussian elimination) work for systems of equations?
5. What is the geometrical interpretation of inconsistent linear systems?

**Tip:** Always check for consistency by verifying all equations after solving for the variables.

2x + 4y - 3z = 0, 22x + 4y - 3z = 0, 2x - y + z = 3, 3x - 2y + 2z = 5

Explore how to solve a system of linear equations with no solution. This problem involves analyzing inconsistency in a system of four equations with three variables using substitution and elimination methods. Detailed steps provided for students in Grades 10-12.

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