The equations $x + 2y + 3z = 1,$ $2x + y + 3z = 2,$ and $5x + 5y + 9z = 4$ have:

If the system of equations $x+4y-z=\lambda$,$7x+9y+\mu z=-3$,and $5x+y+2z=-1$ has infinitely many solutions,then $(2\mu+3\lambda)$ is equal to:

If $(\alpha, \beta, \gamma)$ is the solution of the system of simultaneous linear equations given by $3x + 4y - 5z = -6$,$2x + 3y - 4z = -7$,and $4x - 2y + z = 9$,then find the value of $\alpha + 3\beta - 2\gamma$.

Solve the system of linear equations using the matrix method: $x-y+z=4$,$2x+y-3z=0$,and $x+y+z=2$.

The linear system of equations $\begin{cases} 8x - 3y - 5z = 0 \\ 5x - 8y + 3z = 0 \\ 3x + 5y - 8z = 0 \end{cases}$ has

If ${a_1}x + {b_1}y + {c_1}z = 0, {a_2}x + {b_2}y + {c_2}z = 0, {a_3}x + {b_3}y + {c_3}z = 0$ and $\left| \begin{matrix} {a_1} & {b_1} & {c_1} \\ {a_2} & {b_2} & {c_2} \\ {a_3} & {b_3} & {c_3} \end{matrix} \right| = 0$,then the given system has