Solve the following equations by reduction method :
`2x + 3y = 9, y - x = -2`

Answer

To solve the equations \(2x + 3y = 9\) and \(y - x = -2\) using the reduction method, we will follow these steps:

### Step 1: Write the equations in standard form
The given equations are:
1. \(2x + 3y = 9\) (Equation 1)
2. \(y - x = -2\) (Equation 2)

We can rewrite Equation 2 as:
\[ -x + y = -2 \implies x - y = 2 \quad \text{(Equation 2')} \]

### Step 2: Convert the equations into matrix form
The system of equations can be represented in matrix form as:
\[ \begin{bmatrix} 2 & 3 \\ 1 & -1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 9 \\ 2 \end{bmatrix} \]

### Step 3: Perform row operations
We will perform row operations to eliminate one of the variables. Let's eliminate \(x\) from the second equation.

1. Multiply the second equation (Equation 2') by 2:
\[ 2(x - y) = 2 \cdot 2 \implies 2x - 2y = 4 \quad \text{(Equation 3)} \]

2. Now, we will subtract Equation 3 from Equation 1:
\[ (2x + 3y) - (2x - 2y) = 9 - 4 \]
This simplifies to:
\[ 5y = 5 \]

### Step 4: Solve for \(y\)
Now, divide both sides by 5:
\[ y = 1 \]

### Step 5: Substitute \(y\) back to find \(x\)
Now that we have \(y\), we can substitute it back into one of the original equations to find \(x\). We can use Equation 2:
\[ y - x = -2 \implies 1 - x = -2 \]
Solving for \(x\):
\[ -x = -2 - 1 \implies -x = -3 \implies x = 3 \]

### Step 6: Write the solution
The solution to the system of equations is:
\[ x = 3, \quad y = 1 \]

### Summary of the Solution
The values of \(x\) and \(y\) that satisfy the equations are:
\[ (x, y) = (3, 1) \]