The system of linear equations 3x - 2y - kz = 10, 2x - 4y - 2z = 6, x + 2y - z = 5m is inconsistent

Q: Wait, why do we use Cramer's Rule? Can't we just use Gaussian elimination?

You absolutely can! Gaussian elimination is a powerful, universal tool. However, for a $3 \times 3$ system with parameters, Cramer's Rule is often more elegant and direct. It allows us to isolate the conditions for inconsistency ($\Delta = 0$ and $\Delta_i \neq 0$) without the tedious row-reduction steps that might introduce algebraic errors.

Q: What if $\Delta = 0$ and $\Delta_x = 0$ and $\Delta_y = 0$ and $\Delta_z = 0$? Does that mean it's consistent?

That is a fantastic question! If all determinants are zero, the system is not necessarily inconsistent; it could have infinitely many solutions. Inconsistency specifically requires $\Delta = 0$ AND at least one of the modified determinants to be non-zero. If all are zero, you have a dependent system, not an inconsistent one.

Q: Why do we only check $\Delta_x$? Do we need to check $\Delta_y$ and $\Delta_z$ too?

In many textbook problems, checking one modified determinant is sufficient to establish the condition for inconsistency. If $\Delta_x \neq 0$, the system is already proven inconsistent. If you found $\Delta_x = 0$, you would then check $\Delta_y$ or $\Delta_z$. In this specific problem, the condition $m \neq \frac{4}{5}$ derived from $\Delta_x \neq 0$ is the key constraint.

Question

The system of linear equations $3x - 2y - kz = 10$, $2x - 4y - 2z = 6$, $x + 2y - z = 5m$ is inconsistent if :

Options:

(A) $k = 3, m = \frac{4}{5}$

(B) $k 
eq 3, m \in R$

(C) $k 
eq 3, m 
eq \frac{4}{5}$

(D) $k = 3, m 
eq \frac{4}{5}$

Sigma Percentile · Accepted Answer

### The Geometry of Inconsistency: A Journey into Linear Systems

Welcome, fellow traveler of the mathematical landscape. Today, we are going to peel back the layers of a classic JEE Advanced problem. We are looking at a system of three linear equations with three variables, but with a twist: two mysterious parameters, $k$ and $m$, are lurking within.

Our mission is to find the conditions under which this system becomes **inconsistent**.

### Phase 1: Visualizing the Planes

Imagine you are standing in a three-dimensional room. Each of our three equations represents a flat plane in this space. Normally, three planes intersect at a single point, like the corner of a room where two walls and the ceiling meet.

But what if they don't? What if they are parallel, or arranged like the pages of an open book, never meeting at a single point? That is the essence of an **inconsistent system**.

Mathematically, we say there is no solution. To detect this, we turn to the elegant machinery of Cramer's Rule.

### Phase 2: The Cramer's Rule Toolkit

Cramer's Rule gives us a precise diagnostic test. For a system to be inconsistent, two conditions must be satisfied simultaneously:

1. The main coefficient determinant, $\Delta$, must be exactly zero. This tells us the system is 'degenerate'—the planes are not independent in a way that allows for a unique intersection.

2. At least one of the modified determinants—$\Delta_x, \Delta_y, 	ext{ or } \Delta_z$—must be non-zero. This ensures that the planes are not just dependent (which would lead to infinitely many solutions), but actually contradictory.

### Phase 3: The Calculation of $\Delta$

Let's extract the coefficients of $x, y, z$ to form our matrix:

$$\Delta = \begin{vmatrix} 3 & -2 & -k \ 2 & -4 & -2 \ 1 & 2 & -1 \end{vmatrix}$$

Expanding along the first row, we proceed with caution. Signs are the silent killers in JEE exams! We have:

$$\Delta = 3[(-4)(-1) - (2)(-2)] - (-2)[(2)(-1) - (1)(-2)] + (-k)[(2)(2) - (1)(-4)]$$

Simplifying the minors:

$$\Delta = 3(4 + 4) + 2(-2 + 2) - k(4 + 4)$$

$$\Delta = 3(8) + 2(0) - 8k = 24 - 8k$$

For the system to be inconsistent, we set $\Delta = 0$:

$$24 - 8k = 0 \ \Rightarrow 8k = 24 \ \Rightarrow k = 3$$

We have cracked the first part of the code! $k$ must be $3$.

### Phase 4: The Non-Zero Condition for $\Delta_x$

Now, we must ensure the system is truly inconsistent by checking $\Delta_x$. We replace the first column of our determinant with the constants $(10, 6, 5m)$ and set $k=3$:

$$\Delta_x = \begin{vmatrix} 10 & -2 & -3 \ 6 & -4 & -2 \ 5m & 2 & -1 \end{vmatrix}$$

Expanding this again along the first row:

$$\Delta_x = 10[(-4)(-1) - (2)(-2)] - (-2)[(6)(-1) - (5m)(-2)] + (-3)[(6)(2) - (5m)(-4)]$$

$$\Delta_x = 10(4 + 4) + 2(-6 + 10m) - 3(12 + 20m)$$

$$\Delta_x = 10(8) - 12 + 20m - 36 - 60m$$

$$\Delta_x = 80 - 48 - 40m = 32 - 40m$$

For inconsistency, we require $\Delta_x 
eq 0$:

$$32 - 40m 
eq 0 \ \Rightarrow 40m 
eq 32 \ \Rightarrow m 
eq \frac{32}{40} \ \Rightarrow m 
eq \frac{4}{5}$$

### The Final Verdict

By systematically applying Cramer's Rule, we have uncovered the truth. The system is inconsistent if and only if **$k = 3$** and **$m 
eq \frac{4}{5}$**.

This perfectly aligns with our requirements. Remember, the math is not just about the numbers; it is about understanding the geometric constraints of the space you are working in. Keep practicing, stay curious, and never let a negative sign intimidate you!

Animated Solution for Mathematics - Matrices and Determinants: The system of linear equations $3 x - 2 y - k z = 10$ , $2 x - 4 y - 2 z = 6$ , $x + 2 y - z = 5 m$ is inconsistent if :

Select Answer:

Visualized Solution

System of Linear Equations

Condition for Inconsistency

Setting up the Coefficient Determinant $Δ$

Expanding $Δ$ Along the First Row

Simplifying the Expression for $Δ$

Solving for $k$

Setting up $Δ_{x}$

Expanding $Δ_{x}$ Along the First Row

Simplifying the Expression for $Δ_{x}$

Applying the Non-Zero Condition for $Δ_{x}$

Final Conclusion

The Sigma Insight: Solution of System of Linear Equations (Matrix Method and Cramer's Rule)

The Geometry of Inconsistency

Phase 1

Phase 2

Phase 3

Phase 4

The Final Verdict

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Animated Solution for Mathematics - Matrices and Determinants: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The system of linear equations 3x−2y−kz=10, 2x−4y−2z=6, x+2y−z=5m is inconsistent if :

Select Answer:

Visualized Solution

System of Linear Equations

Condition for Inconsistency

Setting up the Coefficient Determinant .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Δ

Expanding .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Δ Along the First Row

Simplifying the Expression for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Δ

Solving for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } k

Setting up .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Δx​

Expanding .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Δx​ Along the First Row

Simplifying the Expression for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Δx​

Applying the Non-Zero Condition for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Δx​