We have the system of linear equations:

1. $3x + 2y = 22$
2. $9x - 8y = -4$

To solve this system, we can use either substitution or elimination. Let's use the elimination method.

Step 1: Multiply the first equation by 4 and the second equation by 1 (so that the coefficients of $y$ in both equations will cancel out).

$4(3x + 2y) = 4(22) \implies 12x + 8y = 88$ $1(9x - 8y) = 1(-4) \implies 9x - 8y = -4$

Now, we have:

1. $12x + 8y = 88$
2. $9x - 8y = -4$

Step 2: Add the two equations to eliminate $y$.

$(12x + 8y) + (9x - 8y) = 88 + (-4)$ $12x + 9x = 88 - 4$ $21x = 84$

Step 3: Solve for $x$.

$x = \frac{84}{21} = 4$

Step 4: Substitute $x = 4$ into the first equation to solve for $y$.

$3(4) + 2y = 22$ $12 + 2y = 22$ $2y = 22 - 12$ $2y = 10$ $y = \frac{10}{2} = 5$

Final Solution:

$x = 4, \quad y = 5$

Would you like a more detailed explanation of any part of the solution, or do you have other questions? Here's a list of related topics you might be interested in:

1. How to apply the substitution method to solve systems of equations.
2. Solving systems of equations with three variables.
3. Using matrices to solve linear systems.
4. Graphing systems of equations to find their solution visually.
5. Real-world applications of solving systems of equations.

Tip: When using the elimination method, always aim to make the coefficients of one variable match in both equations so they can cancel out when added or subtracted!