Solve the pair of linear equations $x+2y+2=0$ and $3x+2y-2=0$ graphically. Shade the triangular region represented by the given equations and the $y-$axis. Also find the area of this region.

Solve the following pair of linear equations graphically:

$x+3y=6, 2x-3y=12$

Also find the area of triangle formed by the lines representing the given equations with the $y-$axis.

Solve the pair of linear equations $x+y=2, 2x-y=1$ graphically. Write an equation of a line passing through the point of intersection of the given pair of linear equations. How many such lines can you find?

Solve the following pair of linear equations by substitution method:

$$\begin{gathered} 2x=5y+12 \\ 3x-2y+4=0 \end{gathered}$$

Solve the following pair of linear equations by substitution method:

$$\begin{gathered} x+y=7 \\ 2x-3y=11 \end{gathered}$$

Solve the following pair of linear equations by elimination method:

$$\begin{gathered} 3x-5y=4 \\ 2y+7=9x \end{gathered}$$

Solve the following pair of linear equations by elimination method:

$$\begin{gathered} 4x-3y-8=0 \\ 6x-y-\frac{29}{3}=0 \end{gathered}$$

Check whether the following pair of linear equations has unique solution or not? If so, solve it by cross-multiplication method:

$23x-17y+11=0$ and $31x+13y-57=0$

If the pair of linear equations $3x-5y=11$ and $\left(5a+b\right)x-\left(7a-b\right)y=4\left(3a-2b\right)+1$ have infinitely many solutions, then find the values of $a$ and $b$.