Use the Addition Method to solve. 
2x-y=1
3x+2y=12

To solve the system of equations using the **addition method** (also known as the elimination method), we aim to eliminate one variable by adding the equations together. Here are the steps:

### System of equations:
\[
\begin{align*}
1. & \quad 2x - y = 1 \\
2. & \quad 3x + 2y = 12
\end{align*}
\]

### Step 1: Align the equations and multiply (if necessary) to eliminate one variable.
We want to eliminate \( y \). The coefficients of \( y \) in the two equations are \( -1 \) (from \( 2x - y = 1 \)) and \( 2 \) (from \( 3x + 2y = 12 \)). To eliminate \( y \), we can multiply the first equation by 2.

\[
2(2x - y) = 2(1) \quad \Rightarrow \quad 4x - 2y = 2
\]

Now, the system of equations is:
\[
\begin{align*}
1'. & \quad 4x - 2y = 2 \\
2.  & \quad 3x + 2y = 12
\end{align*}
\]

### Step 2: Add the two equations to eliminate \( y \).
Now add the two equations:

\[
(4x - 2y) + (3x + 2y) = 2 + 12
\]

Simplifying:

\[
4x + 3x = 14 \quad \Rightarrow \quad 7x = 14
\]

### Step 3: Solve for \( x \).
\[
x = \frac{14}{7} = 2
\]

### Step 4: Substitute \( x = 2 \) back into one of the original equations.
Let's substitute \( x = 2 \) into the first equation \( 2x - y = 1 \).

\[
2(2) - y = 1 \quad \Rightarrow \quad 4 - y = 1
\]

Solving for \( y \):

\[
-y = 1 - 4 \quad \Rightarrow \quad -y = -3 \quad \Rightarrow \quad y = 3
\]

### Final Answer:
The solution to the system is:
\[
x = 2, \quad y = 3
\]

Would you like more details or have any questions?

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Here are 5 related questions:
1. How would you solve the system using substitution instead of the addition method?
2. What happens if both equations are already balanced to eliminate a variable?
3. How can you verify the solution \( x = 2, y = 3 \) is correct?
4. How does the addition method compare to graphing the system of equations?
5. What would happen if you tried to eliminate \( x \) instead of \( y \)?

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

Use the Addition Method to solve. 2x - y = 1, 3x + 2y = 12

Learn how to solve the system of equations 2x - y = 1 and 3x + 2y = 12 using the Addition Method (Elimination Method). This step-by-step guide simplifies algebraic concepts and eliminates variables to find the solution. Suitable for students in Grades 8-10.

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