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Problem 23: For which value of $\lambda$ the given system has infinitely many solutions

Problem 23: For which value of $\lambda$ the given system has infinitely many solutions

\begin{align*} 2x+3y+4z=13,\\ 5x+7y+7z=26,\\ 9x+13y+15z = 13\lambda. \end{align*}

Solution:

Consider the augmented matrix of this system and apply row operations.

\[\left[\begin{array}{c|c} A & b \end{array} \right] = \begin{align*} \left[\begin{array}{rrr|r} 2 & 3 & 4 & 13 \\ 5 & 7 & 7 & 26 \\ 9 & 13 & 15 & 13\lambda \\ \end{array}\right] \end{align*}\] \[\xrightarrow[R3\rightarrow R_3-R_2-2R_1]{R2\rightarrow R_2-2R_1} \left[\begin{array}{rrr|r} 2 & 3 & 4 & 13 \\ 1 & 1 & -1& 0 \\ 0 & 0 & 0 & 13\lambda -52 \\ \end{array}\right]\]

for infinite solutions Rank[A]=Rank[A,b] = 2. So, \begin{align*} 13\lambda -52 =0 \Rightarrow \lambda = 4. \end{align*}

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