To solve the system of equations using the elimination method, let’s go through each step carefully:

Given equations:

1. $3x + 2y = 3$
2. $2x + 3y = 2$

Step 1: Multiply to Align Coefficients

To eliminate $x$ or $y$, we can make the coefficients of $x$ in both equations the same. We will multiply the first equation by 2 and the second equation by 3:

1. $2 \times (3x + 2y) = 2 \times 3 \Rightarrow 6x + 4y = 6$
2. $3 \times (2x + 3y) = 3 \times 2 \Rightarrow 6x + 9y = 6$

This results in the following system:

1. $6x + 4y = 6$
2. $6x + 9y = 6$

Step 2: Subtract to Eliminate $x$

Now, subtract the first equation from the second equation to eliminate $x$:

$(6x + 9y) - (6x + 4y) = 6 - 6$

This simplifies to:

$5y = 0$

So, $y = 0$.

Step 3: Substitute $y = 0$ Back into One of the Original Equations

Now, substitute $y = 0$ into the first equation:

$3x + 2(0) = 3$

This simplifies to:

$3x = 3 \Rightarrow x = 1$

Solution

The solution to the system is $(x, y) = (1, 0)$.

Would you like further details or have any questions about this solution?

Here are five related questions to deepen your understanding:

1. How would the solution change if we used substitution instead of elimination?
2. What is the geometrical interpretation of this solution on the coordinate plane?
3. How can you verify the solution by plugging the values back into the original equations?
4. What are some potential errors to watch out for when using the elimination method?
5. Could a system of equations ever have no solutions or infinite solutions? How would we detect that?

Tip: When using the elimination method, carefully align terms and check arithmetic to avoid errors, especially when working with larger coefficients.