JEE MAIN - Mathematics (2024 - 5th April Evening Shift - No. 9)
The values of $$m, n$$, for which the system of equations
$$\begin{aligned} & x+y+z=4, \\ & 2 x+5 y+5 z=17, \\ & x+2 y+\mathrm{m} z=\mathrm{n} \end{aligned}$$
has infinitely many solutions, satisfy the equation :
Explanation
The given system of linear equations can be represented as,
$$\begin{aligned} & \left(\begin{array}{ccc|c} 1 & 1 & 1 & 4 \\ 2 & 5 & 5 & 17 \\ 1 & 2 & m & n \end{array}\right) \\ & \sim\left(\begin{array}{ccc|c} 1 & 1 & 1 & 4 \\ 0 & 3 & 3 & 9 \\ 0 & 1 & m-1 & n-4 \end{array}\right) \\ & \sim\left(\begin{array}{ccc|c} 1 & 1 & 1 & 4 \\ 0 & 1 & 1 & 3 \\ 0 & 0 & m-2 & n-7 \end{array}\right) \end{aligned}$$
$$\because$$ System of equations has infinitely many solutions
$$\therefore m=2 \& n=7$$
Which satisfy equation given in option (1).
$$\text { (i.e., } 2^2+7^2-14=39 \text { ) }$$
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