The following system of linear equations 3x + 3y + 2z = 9, 3x + 2y + 2z = 9, x - y + 4z = 8 has: (1)

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

You absolutely can! Gaussian elimination is a fantastic tool. However, Cramer's Rule is often faster for a $3 \times 3$ system when you just need to determine the *nature* of the solution (unique vs. infinite vs. none). In this case, calculating the determinant $\Delta$ quickly tells us if the solution is unique. If $\Delta \neq 0$, we know immediately that a unique solution exists, which saves time before we even start solving for $x, y,$ and $z$.

Q: Why did we subtract the second equation from the first? Is that always allowed?

Yes, it is a fundamental property of linear systems! You can perform row operations—like subtracting one equation from another—without changing the solution set. We chose to subtract the second equation from the first because the coefficients of $x$ and $z$ were identical ($3x$ and $2z$). This caused them to cancel out perfectly, leaving us with a simple equation for $y$. It is a 'shortcut' that turns a complex 3D problem into a much simpler 2D one.

Q: What if $\Delta$ had turned out to be zero? How would I know if it has infinitely many solutions or no solution?

That is a great question! If $\Delta = 0$, the system is either inconsistent (no solution) or dependent (infinitely many solutions). To distinguish between them, you would need to calculate the other determinants ($\Delta_x, \Delta_y, \Delta_z$) or use row reduction (Gaussian elimination) to see if you get a contradiction like $0 = 5$ (no solution) or an identity like $0 = 0$ (infinitely many solutions). Always stay calm and proceed step-by-step!

Question

The following system of linear equations $3x + 3y + 2z = 9, 3x + 2y + 2z = 9, x - y + 4z = 8$ has:

Options:

(A) does not have any solution

(B) has a unique solution

(C) has a solution $(\alpha, \beta, \gamma)$ satisfying $\alpha + \beta^2 + \gamma^3 = 12$

(D) has infinitely many solutions

Sigma Percentile · Accepted Answer

### The Geometry of Intersecting Planes

Imagine you are standing in a vast, three-dimensional room. You have three large, flat sheets of glass—these are our planes.

The equations $3x + 3y + 2z = 9$, $3x + 2y + 2z = 9$, and $x - y + 4z = 8$ represent these planes. Our mission is to determine how these three planes interact.

Do they collide at a single, precise point in space? Do they meet along a common line? Or do they perhaps never meet at all?

### The Heartbeat of the System: The Determinant

To unlock the secret of these planes, we turn to the coefficient matrix. We extract the coefficients of $x, y,$ and $z$ to form our matrix:

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

This determinant, $\Delta$, is the heartbeat of our system. If $\Delta 
eq 0$, the planes are guaranteed to intersect at exactly one unique point.

Let's calculate it by expanding along the first row:

$$\Delta = 3(8 - (-2)) - 3(12 - 2) + 2(-3 - 2)$$

$$\Delta = 3(10) - 3(10) + 2(-5)$$

$$\Delta = 30 - 30 - 10 = -10$$

Since $\Delta = -10$, and $-10 
eq 0$, we have mathematically confirmed that the system has a **unique solution**. The planes meet at one single point in space.

### The Elegance of Elimination

Now that we know a unique solution exists, we need to find it. While we could use Cramer's Rule for all variables, there is a much more elegant path.

Look at the first two equations:

1) $3x + 3y + 2z = 9$

2) $3x + 2y + 2z = 9$

If we subtract equation (2) from equation (1), the $3x$ and $2z$ terms vanish into thin air!

$$(3x + 3y + 2z) - (3x + 2y + 2z) = 9 - 9$$

$$y = 0$$

Just like that, we have our first coordinate! With $y = 0$, our system collapses into a simple 2D problem.

Substituting $y = 0$ into equations (2) and (3):

From (2): $3x + 2z = 9$

From (3): $x + 4z = 8 \implies x = 8 - 4z$

Substituting $x = 8 - 4z$ into $3x + 2z = 9$ gives us:

$$3(8 - 4z) + 2z = 9$$

$$24 - 12z + 2z = 9$$

$$-10z = -15 \implies z = 1.5$$

Finally, finding $x$ is a breeze:

$$x = 8 - 4(1.5) = 8 - 6 = 2$$

Our unique intersection point is $(\alpha, \beta, \gamma) = (2, 0, 1.5)$.

### Debunking the Trap

In JEE Advanced, never assume the first option you find is the only one to check. Option (3) suggests a condition: $\alpha + \beta^2 + \gamma^3 = 12$.

Let's test our point $(2, 0, 1.5)$:

$$2 + 0^2 + (1.5)^3 = 2 + 0 + 3.375 = 5.375$$

Since $5.375 
eq 12$, option (3) is clearly a trap. We have successfully navigated the complexity, verified our result, and confirmed that the system has a unique solution.

Keep this level of rigor in your practice, and you will master the art of the JEE!

Animated Solution for Mathematics - Matrices and Determinants: The following system of linear equations $3 x + 3 y + 2 z = 9, 3 x + 2 y + 2 z = 9, x - y + 4 z = 8$ has:

Select Answer:

Visualized Solution

The System of Equations

Cramer's Rule Strategy

Setting up $Δ$

Expanding $Δ$

Result of $Δ$

Finding the Exact Solution

The Elimination Shortcut

Substituting $y = 0$

Solving for $x$ and $z$

Finding $z$

Finding $x$

The Unique Solution Point

Verifying Option 3

Final Conclusion

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

The Geometry of Intersecting Planes

The Heartbeat of the System

The Elegance of Elimination

Debunking the Trap

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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 following system of linear equations 3x+3y+2z=9,3x+2y+2z=9,x−y+4z=8 has:

Select Answer:

Visualized Solution

The System of Equations

Cramer's Rule Strategy

Setting up .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

Result of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Δ

Finding the Exact Solution

The Elimination Shortcut

Substituting .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } y=0

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

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

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