The following system of equations $3x - 7y + 5z = 3$,$3x + y + 5z = 7$,and $2x + 3y + 5z = 5$ is:

In a legislative assembly election,a political group hired a public relations firm to promote its candidate in three ways: telephone,house calls,and letters. The cost per contact (in paise) is given in matrix $A$ as $A = \begin{bmatrix} 40 \\ 100 \\ 50 \end{bmatrix} \begin{matrix} \text{Telephone} \\ \text{Housecall} \\ \text{Letter} \end{matrix}$. The number of contacts of each type made in two cities $X$ and $Y$ is given by $B = \begin{bmatrix} 1000 & 500 & 5000 \\ 3000 & 1000 & 10000 \end{bmatrix} \begin{matrix} \text{Telephone} & \text{Housecall} & \text{Letter} \\ \to X \\ \to Y \end{matrix}$. Find the total amount spent by the group in the two cities $X$ and $Y$.

If the system of linear equations $x_1 + 2x_2 + 3x_3 = 6$,$x_1 + 3x_2 + 5x_3 = 9$,and $2x_1 + 5x_2 + ax_3 = b$ is consistent and has an infinite number of solutions,then:

If $x = a$,$y = b$,$z = c$ is a solution of the system of linear equations $x + 8y + 7z = 0$,$9x + 2y + 3z = 0$,and $x + y + z = 0$ such that the point $(a, b, c)$ lies on the plane $x + 2y + z = 6$,then $2a + b + c$ equals

For a matrix $A=\begin{bmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 2 & 1 \end{bmatrix}$,if $U_{1}, U_{2}$,and $U_{3}$ are $3 \times 1$ column matrices satisfying $A U_{1}=\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$,$A U_{2}=\begin{bmatrix} 2 \\ 3 \\ 0 \end{bmatrix}$,$A U_{3}=\begin{bmatrix} 2 \\ 3 \\ 1 \end{bmatrix}$,and $U$ is a $3 \times 3$ matrix whose columns are $U_{1}, U_{2}$,and $U_{3}$,then the sum of the elements of $U^{-1}$ is:

Let $\lambda, \mu \in R$. If the system of equations
$3x + 5y + \lambda z = 3$
$7x + 11y - 9z = 2$
$97x + 155y - 189z = \mu$
has infinitely many solutions,then $\mu + 2\lambda$ is equal to :