If $\frac{x+y}{x y}=2$ and $\frac{x-y}{x y}=6,$ then $x=\ldots \ldots \ldots \ldots .$

The denominator of a fraction is $6$ more than the numerator. If $1$ is added to the numerator and $3$ is subtracted from the denominator,the fraction becomes $\frac{3}{4}$. Find the fraction.

Solve the following pair of equations: $(a \neq 0, b \neq 0)$
$\frac{2}{a} + \frac{3}{b} = 1$
$\frac{1}{a} + \frac{1}{b} = 1$

Solve the following pair of linear equations by the method of substitution: $2x - y + 3 = 0$ and $y = 2x - \frac{3}{2}$.

Obtain the solution of the following pair of equations by the cross-multiplication method: $ax + by = 1$ and $bx + ay = \frac{2ab}{a^2 + b^2}$.

Graphically,solve the following pair of equations:
$2x + y = 6$
$2x - y + 2 = 0$
Find the ratio of the areas of the two triangles formed by the lines representing these equations with the $x$-axis and the lines with the $y$-axis.