$A$ two-digit number is obtained by either multiplying the sum of the digits by $8$ and adding $1$,or by multiplying the difference of the digits by $13$ and adding $2$. Find the number.

Formulate the following problems as a pair of equations,and hence find their solutions:
Roohi travels $300 \ km$ to her home partly by train and partly by bus. She takes $4 \ hours$ if she travels $60 \ km$ by train and the remaining by bus. If she travels $100 \ km$ by train and the remaining by bus,she takes $10 \ minutes$ longer. Find the speed of the train and the bus separately.

If the following pair of equations does not have any solution,then find the value of $k$:
$3x - 4y + 7 = 0$
$kx + 3y - 5 = 0$

Solve the following pair of linear equations: $2(3u - v) = 5uv$ and $2(u + 3v) = 5uv$.

Places $A$ and $B$ are $100 \ km$ apart on a highway. One car starts from $A$ and another from $B$ at the same time. If the cars travel in the same direction at different speeds,they meet in $5 \ hours$. If they travel towards each other,they meet in $1 \ hour$. What are the speeds of the two cars?

Solve the following pair of equations by the method of cross multiplication:
$ax + by - a + b = 0$
$bx - ay - a - b = 0$