Solve the system by the addition method.
3 x minus 2 y3x−2y
​=
negative 9−9
2 x plus 3 y2x+3y
​=
negative 6−6
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Part 1
Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.
A.
The solution set is StartSet nothing EndSet .enter your response here. ​(Type an ordered​ pair.)
B.
There are infinitely many solutions.
C.
There is no solution.

To solve the system of equations using the addition method (also known as the elimination method), we first write the system of equations clearly:

1. \( 3x - 2y = -9 \)
2. \( 2x + 3y = -6 \)

Next, we need to eliminate one of the variables by adding the two equations together. We can do this by multiplying each equation by a suitable number so that the coefficients of one of the variables are opposites. 

Let's eliminate \( y \) by making the coefficients of \( y \) in both equations equal but opposite. To do this, we multiply the first equation by 3 and the second equation by 2:

\[
\begin{align*}
3(3x - 2y) &= 3(-9) \\
2(2x + 3y) &= 2(-6)
\end{align*}
\]

Simplifying these, we get:

\[
\begin{align*}
9x - 6y &= -27 \quad \text{(Equation 3)} \\
4x + 6y &= -12 \quad \text{(Equation 4)}
\end{align*}
\]

Now we add Equation 3 and Equation 4 together:

\[
(9x - 6y) + (4x + 6y) = -27 + (-12)
\]

Simplifying the left side, we see that the \( y \) terms cancel out:

\[
9x + 4x = -27 - 12
\]

This simplifies to:

\[
13x = -39
\]

Solving for \( x \):

\[
x = \frac{-39}{13} = -3
\]

Now that we have \( x \), we substitute \( x = -3 \) back into one of the original equations to find \( y \). We'll use the first equation:

\[
3x - 2y = -9
\]

Substitute \( x = -3 \):

\[
3(-3) - 2y = -9
\]

Simplify:

\[
-9 - 2y = -9
\]

Add 9 to both sides:

\[
-2y = 0
\]

Solve for \( y \):

\[
y = 0
\]

So, the solution to the system is \( (x, y) = (-3, 0) \). 

Thus, the correct choice is:

**A. The solution set is \( \{(-3, 0)\} \).**

Do you want details or have any questions?

Here are 5 questions you can ask next:
1. How do you solve a system of equations using substitution?
2. Can you explain how to solve a quadratic equation?
3. How do you graph the solution to a system of equations?
4. What is the difference between consistent and inconsistent systems?
5. How do you solve a system of equations using matrices?

**Tip:** Always check your solution by substituting back into the original equations to ensure accuracy.

Learn how to solve a system of equations using the addition (elimination) method. Step-by-step solution to the system: 3x - 2y = -9 and 2x + 3y = -6, including detailed algebraic manipulation and verification of solution (x, y) = (-3, 0). Suitable for high school students and beyond.

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