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

Question

The values of \(m, n\), for which the system of equations

\(\mathrm{x}+\mathrm{y}+\mathrm{z}=4\),

\(\mathrm{2 x+5 y+5 z=17}\),

\(\mathrm{x}+2 \mathrm{y}+\mathrm{mz}=\mathrm{n}\)

has infinitely many solutions, satisfy the equation :

(1) \(m^{2}+n^{2}-m-n=46\)

(2) \(m^{2}+n^{2}+m+n=64\)

(3) \(m^{2}+n^{2}+m n=68\)

(4) \(m^{2}+n^{2}-m n=39\)

Answer 1

Correct option is (4) \(m^{2}+n^{2}-m n=39\)

\(\mathrm{D}=\left|\begin{array}{ccc}1 & 1 & 1 \\ 2 & 5 & 5 \\ 1 & 2 & \mathrm{~m}\end{array}\right|=0 \Rightarrow \mathrm{m}=2\)

\(\mathrm{D}_{3}=\left|\begin{array}{ccc}1 & 1 & 4 \\ 2 & 5 & 17 \\ 1 & 2 & \mathrm{n}\end{array}\right|=0 \Rightarrow \mathrm{n}=7\)

\(\Rightarrow m^2 + n^2 - mn =39\)