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?

For which values of $a$ and $b$ does the following pair of linear equations have an infinite number of solutions?

$2x+3y=7$

$\left(a-b\right)x+\left(a+b\right)y=3a+b-2$

Solve the following pair of equations  by reducing them to a pair of linear equation:

$\frac{10}{x+y}+\frac{2}{x-y}=4 ; \frac{15}{x+y}-\frac{5}{x-y}=-2$

The area of a rectangle gets reduced by $9$ square units, if its length is reduced by $5$ units and breadth is increased by $3$ units.  If the length is increased by $3$ units and breadth by $2$ units, the area is increased by $67$ square units. Find the length and breadth of the rectangle .

Consider the following pairs of linear equation:

$$\begin{gathered} \left(i\right) 3x+2y=5 ; 2x+3y=5 \\ \left(ii\right) 2x-3y=7; 2x-3y=8 \end{gathered}$$

Choose the correct alternative.

Half the perimeter of a rectangular garden, whose length is $4 m$ more than its width, is $36 m$. Find the dimensions of the garden.

Solve the pair of equations by reducing them to a pair of linear equations: 

$\frac{5}{x-1}+\frac{1}{y-2}=2$

$\frac{6}{x-1}-\frac{3}{y-2}=1$

Consider the following pair of linear equations:

(i) $2x-3y=8, 4x-6y=9$

(ii) $2x+3y-9=0, 4x+6y-18=0$

$5$ pencils and $7$ pens together cost $₹50$, whereas $7$ pencils and $5$ pens together cost $₹46$. Find the cost of one pencil and that of one pen.