We are given the following system of equations:

1. $4x + 8y = 12$
2. $3x - 8y = 9$

We will use the elimination method to solve this system.

Step 1: Eliminate one of the variables

In this case, the coefficients of $y$ are opposites (+8 and -8), so we can eliminate $y$ by adding the two equations together.

$(4x + 8y) + (3x - 8y) = 12 + 9$

Simplify:

$4x + 3x + 8y - 8y = 21$ $7x = 21$

Step 2: Solve for $x$

Now, solve for $x$:

$x = \frac{21}{7}$ $x = 3$

Step 3: Substitute $x = 3$ into one of the original equations

We will substitute $x = 3$ into the first equation $4x + 8y = 12$:

$4(3) + 8y = 12$ $12 + 8y = 12$

Step 4: Solve for $y$

Now, subtract 12 from both sides:

$8y = 0$

Finally, divide by 8:

$y = 0$

Solution:

The solution to the system is:

$x = 3, \quad y = 0$

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Would you like more details or have any questions?

Here are some related questions for further exploration:

1. How would you solve a system with fractions using elimination?
2. Can the elimination method be used with non-linear systems?
3. What is the graphical interpretation of the solution to a system of linear equations?
4. How does the substitution method compare to the elimination method?
5. What would happen if the two lines in this system were parallel?

Tip: When using the elimination method, always look for coefficients that are the same or additive inverses to eliminate a variable easily.