The ratio of incomes of two persons is $9:7$ and the ratio of their expenditures is $4:3$. If each of them manages to save ₹ $2000$ per month,find their monthly incomes.

Solve the following pair of equations by reducing them to a pair of linear equations:
$\frac{4}{x} + 3y = 14$
$\frac{3}{x} - 4y = 23$

Form the pair of linear equations for the following problem,and find their solutions (if they exist) by the elimination method:
If we add $1$ to the numerator and subtract $1$ from the denominator,a fraction reduces to $1$. It becomes $\frac{1}{2}$ if we only add $1$ to the denominator. What is the fraction?

Two rails are represented by the equations $x+2y-4=0$ and $2x+4y-12=0$. Represent this situation geometrically.

On comparing the ratios $\frac{a_{1}}{a_{2}}, \frac{b_{1}}{b_{2}}$ and $\frac{c_{1}}{c_{2}}$,find out whether the following pair of linear equations are consistent or inconsistent:
$\frac{4}{3} x + 2 y = 8$; $2 x + 3 y = 12$

Formulate the following problems as a pair of equations,and hence find their solutions:
$2$ women and $5$ men can together finish an embroidery work in $4$ days,while $3$ women and $6$ men can finish it in $3$ days. Find the time taken by $1$ woman alone to finish the work,and also that taken by $1$ man alone.