If the system of equations: x + y + z = 5 x + 2y + 3z = 9 x + 3y + lambda z = mu has infinitely many solutions, then the value of lambda + mu is:

The correct answer is Option 2. Full mathematical solution steps and hints are accessible in the interactive StemCET workspace.

Maths Q3 – JEE-MAIN 2026

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Home Maths MatricesQuestion #287

If the system of equations:

x + y + z = 5

x + 2 y + 3 z = 9

x + 3 y + λ z = μ

has infinitely many solutions, then the value of

λ + μ

is:

Clue 1 of 3: Conditions for Infinitely Many Solutions

For a system of linear equations to have infinitely many solutions, the rank of the coefficient matrix must be equal to the rank of the augmented matrix, and this rank must be less than the number of variables.

🧠 Full Solution Path

Step 1: Represent as Augmented Matrix and Perform Row Operations✦ Active

The given system of equations can be written in augmented matrix form:

\begin{array}{cccc} 1 & 1 & 1 & 5 \\ 1 & 2 & 3 & 9 \\ 1 & 3 & λ & μ \end{array}

Perform row operations: $R_{2} \to R_{2} - R_{1}$ and $R_{3} \to R_{3} - R_{1}$ .

\begin{array}{cccc} 1 & 1 & 1 & 5 \\ 0 & 1 & 2 & 4 \\ 0 & 2 & λ - 1 & μ - 5 \end{array}

Then, perform $R_{3} \to R_{3} - 2 R_{2}$ .

\begin{array}{cccc} 1 & 1 & 1 & 5 \\ 0 & 1 & 2 & 4 \\ 0 & 0 & λ - 5 & μ - 13 \end{array}

Step 2: Apply Condition for Infinitely Many Solutions○ Expand

For the system to have infinitely many solutions, the last row of the row-echelon form of the augmented matrix must be entirely zeros. This means both the coefficient and the constant term in the last row must be zero.

Therefore, $λ - 5 = 0$ and $μ - 13 = 0$ .

Solving these equations gives $λ = 5$ and $μ = 13$ .

Step 3: Calculate

λ + μ

○ Expand

The value of $λ + μ$ is $5 + 13 = 18$ .

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Maths Question 3 – JEE-MAIN 2026

Clue 1 of 3: Conditions for Infinitely Many Solutions

🧠 Full Solution Path