The given system of simultaneous equations is:

$1) \ 2x + 3y = 6$ $2) \ 3x - 2y = 22$

Solving the system of equations:

Step 1: Multiply the equations to eliminate one variable

We can eliminate $y$ by multiplying the first equation by 2 and the second equation by 3:

$1) \ 4x + 6y = 12$ $2) \ 9x - 6y = 66$

Step 2: Add the two equations

Adding both equations together eliminates $y$:

$(4x + 6y) + (9x - 6y) = 12 + 66$ $13x = 78$

Step 3: Solve for $x$

$x = \frac{78}{13} = 6$

Step 4: Substitute $x = 6$ into one of the original equations

We substitute into the first equation: $2(6) + 3y = 6$ $12 + 3y = 6$ $3y = 6 - 12$ $3y = -6 \quad \Rightarrow \quad y = \frac{-6}{3} = -2$

Final Solution:

The solution to the system of equations is: $x = 6, \quad y = -2$

Would you like further clarification on any step?

Here are 5 related questions to explore further:

1. What methods can be used to solve systems of equations other than elimination?
2. How would you solve the system if it involved fractions?
3. Can substitution be used to solve the same system of equations? How?
4. What is the geometric interpretation of solving two simultaneous equations?
5. How do we identify whether a system of equations has no solution or infinitely many solutions?

Tip: When solving simultaneous equations, ensure that you're clear on which variable you aim to eliminate first for a smoother process!