Solve the following pair of linear equations by elimination method :
`x+y=5 and 2x-3y=4`.

Answer

To solve the pair of linear equations using the elimination method, we have the following equations:

1. \( x + y = 5 \) (Equation 1)
2. \( 2x - 3y = 4 \) (Equation 2)

### Step 1: Multiply Equation 1 to Align Coefficients
We want to eliminate one of the variables by making the coefficients of \( y \) in both equations equal. We can do this by multiplying Equation 1 by 3:

\[ 3(x + y) = 3(5) \]

This gives us:

\[ 3x + 3y = 15 \quad \text{(Equation 3)} \]

### Step 2: Write Down the New Set of Equations
Now we have:

1. \( 3x + 3y = 15 \) (Equation 3)
2. \( 2x - 3y = 4 \) (Equation 2)

### Step 3: Add the Equations to Eliminate \( y \)
Next, we will add Equation 3 and Equation 2 to eliminate \( y \):

\[ (3x + 3y) + (2x - 3y) = 15 + 4 \]

This simplifies to:

\[ 5x + 0y = 19 \]

So we have:

\[ 5x = 19 \]

### Step 4: Solve for \( x \)
Now, divide both sides by 5 to find \( x \):

\[ x = \frac{19}{5} \]

### Step 5: Substitute \( x \) Back to Find \( y \)
Now that we have \( x \), we can substitute it back into Equation 1 to find \( y \):

\[ x + y = 5 \]

Substituting \( x = \frac{19}{5} \):

\[ \frac{19}{5} + y = 5 \]

To isolate \( y \), subtract \( \frac{19}{5} \) from both sides:

\[ y = 5 - \frac{19}{5} \]

Convert 5 into a fraction with a denominator of 5:

\[ y = \frac{25}{5} - \frac{19}{5} = \frac{6}{5} \]

### Final Solution
Thus, the solution to the equations is:

\[ x = \frac{19}{5}, \quad y = \frac{6}{5} \]