Solve the following pairs of equation by elimination method:
`x + y = 6`
`2x - 3y = 4`.

Answer

To solve the given pair of equations using the elimination method, we follow these steps:

### Step 1: Write down the equations
The given equations are:
1. \( x + y = 6 \) (Equation 1)
2. \( 2x - 3y = 4 \) (Equation 2)

### Step 2: Make the coefficients of one variable the same
We can eliminate \( x \) by manipulating Equation 1. To do this, we can multiply Equation 1 by 2 so that the coefficients of \( x \) in both equations become the same.

\[ 2(x + y) = 2(6) \]
This simplifies to:
\[ 2x + 2y = 12 \quad \text{(Equation 3)} \]

### Step 3: Write the new system of equations
Now we have:
1. \( 2x + 2y = 12 \) (Equation 3)
2. \( 2x - 3y = 4 \) (Equation 2)

### Step 4: Subtract the equations
Next, we will subtract Equation 2 from Equation 3 to eliminate \( x \):

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

This simplifies to:
\[ 2y + 3y = 8 \]
\[ 5y = 8 \]

### Step 5: Solve for \( y \)
Now, we can solve for \( y \):
\[ y = \frac{8}{5} \]

### Step 6: Substitute \( y \) back into one of the original equations
We can substitute \( y = \frac{8}{5} \) back into Equation 1 to find \( x \):
\[ x + \frac{8}{5} = 6 \]

### Step 7: Solve for \( x \)
To isolate \( x \), we subtract \( \frac{8}{5} \) from both sides:
\[ x = 6 - \frac{8}{5} \]

To perform this subtraction, we convert 6 into a fraction with a denominator of 5:
\[ x = \frac{30}{5} - \frac{8}{5} = \frac{30 - 8}{5} = \frac{22}{5} \]

### Step 8: Write the final solution
Thus, the solution to the system of equations is:
\[ x = \frac{22}{5}, \quad y = \frac{8}{5} \]

### Summary of the solution
The values of \( x \) and \( y \) are:
- \( x = \frac{22}{5} \)
- \( y = \frac{8}{5} \)