Solve the following pairs of equations by the method of elimination:
$5x - 3y = 1$
$2x + 5y = 19$

While arranging the students of standard $X$ in rows,if two rows are reduced,then one more student has to be arranged in each of the remaining rows. If three more rows are formed,then one student has to be taken off from each of the previously arranged rows. Find the total number of students in the class.

Solve the following pairs of equations by the method of elimination:
$5(x-2) + 3y = 1$
$3(x+1) + 5(y-3) = 1$

$A$ 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.

Do the equations $4x + 3y - 1 = 5$ and $12x + 9y = 15$ represent a pair of coincident lines? Justify your answer.

Solve the following pairs of equations by the cross-multiplication method:
$4x - 19y + 13 = 0$
$13x - 23y + 19 = 0$