We are given the system of equations:

1. $3x + 2y = 4$
2. $5x - 2y = 12$

We will use the Addition Method (also called the Elimination Method) to solve this system.

Step 1: Add the two equations

To eliminate $y$, notice that the coefficients of $y$ in both equations are $+2$ and $-2$. When we add the two equations, the $y$-terms will cancel out.

$(3x + 2y) + (5x - 2y) = 4 + 12.$

Simplify: $3x + 5x + 2y - 2y = 16,$ $8x = 16.$

Step 2: Solve for $x$

Now, solve for $x$ by dividing both sides by 8: $x = \frac{16}{8} = 2.$

Step 3: Substitute $x = 2$ into one of the original equations

Now that we know $x = 2$, substitute this value into one of the original equations. Let's use the first equation $3x + 2y = 4$.

Substitute $x = 2$: $3(2) + 2y = 4,$ $6 + 2y = 4.$

Step 4: Solve for $y$

Solve for $y$ by subtracting 6 from both sides: $2y = 4 - 6,$ $2y = -2.$

Now, divide both sides by 2: $y = \frac{-2}{2} = -1.$

Final Answer:

The solution to the system of equations is: $x = 2, \quad y = -1.$ So, the solution is the point $(2, -1)$.

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Would you like to see another example or explore a different method? Here are some related questions:

1. How does the Addition Method compare to the Substitution Method?
2. How would the system change if one of the equations had fractions?
3. What happens if the coefficients of $x$ and $y$ are the same but the constants are different?
4. How can we check if the solution is correct by substitution?
5. What would happen if the system of equations had no solution or infinitely many solutions?

Tip: The Addition Method works best when the coefficients of one of the variables are opposites or can be made opposites through multiplication.