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

Question

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 :

Accepted Answer

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}\end{aligned}$$\right) \
\& \sim\left($$\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)
\]

$$\because$$ System of equations has infinitely many solutions

$$\therefore m=2 \& n=7$$

Which satisfy equation given in option (1).

$$\text { (i.e., } 2^2+7^2-14=39 \text { ) }$$

Solutions · 1