Find the co-ordinates of the point of intersection of the straight lines
`2x-3y-7=0 , 3x-4y-13=0`

Answer

To find the coordinates of the point of intersection of the straight lines given by the equations:

1. \( 2x - 3y - 7 = 0 \) (Equation 1)
2. \( 3x - 4y - 13 = 0 \) (Equation 2)

we will follow these steps:

### Step 1: Rewrite the equations
We can keep the equations in their standard form for easier manipulation:

- Equation 1: \( 2x - 3y = 7 \)
- Equation 2: \( 3x - 4y = 13 \)

### Step 2: Eliminate one variable
To eliminate one variable, we can multiply both equations by suitable numbers so that the coefficients of \( x \) in both equations become the same.

Let's multiply Equation 1 by 3 and Equation 2 by 2:

- \( 3(2x - 3y) = 3(7) \) → \( 6x - 9y = 21 \) (Equation 3)
- \( 2(3x - 4y) = 2(13) \) → \( 6x - 8y = 26 \) (Equation 4)

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

\[ (6x - 9y) - (6x - 8y) = 21 - 26 \]

This simplifies to:

\[ -9y + 8y = -5 \]

\[ -y = -5 \]

\[ y = 5 \]

### Step 4: Substitute back to find \( x \)
Now that we have \( y \), we can substitute this value back into either Equation 1 or Equation 2 to find \( x \). We will use Equation 1:

\[ 2x - 3(5) = 7 \]

This simplifies to:

\[ 2x - 15 = 7 \]

Adding 15 to both sides gives:

\[ 2x = 22 \]

Dividing by 2:

\[ x = 11 \]

### Step 5: Write the coordinates of intersection
The coordinates of the point of intersection of the two lines are:

\[ (x, y) = (11, 5) \]

### Final Answer
The coordinates of the point of intersection are \( (11, 5) \).

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