The values of m, n, for which the system of equations x + y + z = 4, 2x + 5y + 5z = 17, x + 2y + mz

Q: Why do we need $D_1, D_2, D_3$ to be zero? Isn't $D=0$ enough for infinite solutions?

That is a fantastic question! $D=0$ only tells us that the system does not have a unique solution. It could mean the system is inconsistent (no solution) or dependent (infinitely many). We need the substituted determinants to be zero to ensure the system is consistent. If $D=0$ but $D_3 \neq 0$, the planes are parallel or arranged such that they never share a common line, leading to no solution.

Q: Can I use Gaussian elimination instead of Cramer's Rule?

Absolutely! In fact, row reduction is often faster and less prone to arithmetic errors. If you perform row operations to get the system into row-echelon form, you can see the dependency directly. Cramer's Rule is just one tool in your belt; use whichever feels more intuitive to you.

Q: What if I get $m=2$ but the options don't match?

Don't panic! If your $m$ is correct, double-check your calculation for $n$. The problem asks for the values of $m$ and $n$ that satisfy a specific equation. Ensure you have correctly substituted your values into the expression $m^2 + n^2 - mn$ and performed the arithmetic carefully. Precision is key in JEE Advanced!

Question

The values of $m, n$, for which the system of equations $x + y + z = 4$, $2x + 5y + 5z = 17$, $x + 2y + mz = n$ has infinitely many solutions, satisfy the equation:

Options:

(A) $m^2 + n^2 - mn = 39$

(B) $m^2 + n^2 - m - n = 46$

(C) $m^2 + n^2 + m + n = 64$

(D) $m^2 + n^2 + mn = 68$

Sigma Percentile · Accepted Answer

### The Harmony of Intersecting Planes

Imagine you are standing in a three-dimensional space. You have three sheets of glass, each representing a linear equation: $x + y + z = 4$, $2x + 5y + 5z = 17$, and $x + 2y + mz = n$.

Usually, these three planes would intersect at a single point, giving us a unique solution for $(x, y, z)$. But today, we are looking for something more elegant.

We want these planes to intersect along an entire line. This is the geometric definition of a system with infinitely many solutions, a state of perfect dependency where the equations are in harmony.

### The Toolkit: Cramer's Rule

To find this harmony, we turn to the power of determinants. For a non-homogeneous system $AX = B$, the condition for a unique solution is $D 
eq 0$.

When we seek infinite solutions, we require the system to be consistent yet dependent. This forces the main determinant, $D$, to vanish: $D = 0$.

However, $D=0$ is not enough, as it could also imply an inconsistent system (no solution). To guarantee consistency, we must also ensure that the substituted determinants $D_1, D_2,$ and $D_3$ are zero.

### Phase 1: The Hunt for $m$

We extract the coefficients of our variables $x, y,$ and $z$ to form our matrix:

$$D = \begin{vmatrix} 1 & 1 & 1 \ 2 & 5 & 5 \ 1 & 2 & m \end{vmatrix}$$

To find $m$, we set $D = 0$ and expand along the first row. Expanding gives us:

$$1(5m - 10) - 1(2m - 5) + 1(4 - 5) = 0$$

Let us simplify this step-by-step. We have $5m - 10 - 2m + 5 - 1 = 0$.

Combining the terms, we get $3m - 6 = 0$, which leads us directly to **$m = 2$**. We have successfully unlocked the first parameter.

### Phase 2: The Hunt for $n$

Now that we know $m=2$, we must find $n$. We invoke the condition $D_3 = 0$.

We replace the third column of our determinant with the constants from the right-hand side of our equations: $4, 17,$ and $n$.

$$D_3 = \begin{vmatrix} 1 & 1 & 4 \ 2 & 5 & 17 \ 1 & 2 & n \end{vmatrix} = 0$$

Expanding this along the first row, we get:

$$1(5n - 34) - 1(2n - 17) + 4(4 - 5) = 0$$

Again, we simplify with care: $5n - 34 - 2n + 17 - 4 = 0$.

Grouping the $n$ terms gives $3n$, and the constants sum to $-21$. Thus, $3n - 21 = 0$, which yields **$n = 7$**.

### The Grand Finale

We have found our values: $m = 2$ and $n = 7$. The problem asks us to verify which equation these values satisfy.

We test the expression $m^2 + n^2 - mn$:

$$2^2 + 7^2 - (2)(7) = 4 + 49 - 14$$

Calculating this, we get $53 - 14 = 39$.

This matches the required result perfectly. You have navigated the geometry of planes, mastered the determinants, and solved for the parameters.

Animated Solution for Mathematics - Matrices and Determinants: The values of $m, n$ , for which the system of equations $x + y + z = 4$ , $2 x + 5 y + 5 z = 17$ , $x + 2 y + m z = n$ has infinitely many solutions, satisfy the equation:

Select Answer:

Visualized Solution

System of Linear Equations

Condition for Infinite Solutions

Setting up Determinant $D$

Expanding Determinant $D$

Solving for $m$

Setting up Determinant $D_{3}$

Expanding Determinant $D_{3}$

Solving for $n$

Evaluating the Options

Final Calculation

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

The Harmony of Intersecting Planes

The Toolkit

Phase 1

Phase 2

The Grand Finale

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Visualized Solution

System of Linear Equations

Condition for Infinite Solutions

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

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

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

Setting up Determinant .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } D3​

Expanding Determinant .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } D3​

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