What is a Consistent Pair of Linear Equations?

In Fig. 1, $ABCD$ is a rectangle. Find the values of $x$ and $y$.

Find $c$ if the system of equations $cx+3y+\left(3-c\right)=0$; $12x+cy-c=0$ has infinitely many solutions.

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.

The system of equations $2x+5y=17$ and $5x+3y=14$ has

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 .

$5$ 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.

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.

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

$2x+3y=7$

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