Find whether the following pair of linear equations has a unique solution. If yes, find the solution.
`7x-4y =49 and 5x-6y=57`

Answer

To determine whether the given pair of linear equations has a unique solution, we will follow these steps:

### Given Equations:
1. \( 7x - 4y = 49 \) (Equation 1)
2. \( 5x - 6y = 57 \) (Equation 2)

### Step 1: Identify Coefficients
From the equations, we can identify the coefficients:
- For Equation 1:
- \( A_1 = 7 \)
- \( B_1 = -4 \)
- \( C_1 = -49 \)

- For Equation 2:
- \( A_2 = 5 \)
- \( B_2 = -6 \)
- \( C_2 = -57 \)

### Step 2: Check the Condition for Unique Solution
The condition for a unique solution is given by:
\[ \frac{A_1}{A_2} \neq \frac{B_1}{B_2} \]

Calculating \( \frac{A_1}{A_2} \):
\[ \frac{A_1}{A_2} = \frac{7}{5} \]

Calculating \( \frac{B_1}{B_2} \):
\[ \frac{B_1}{B_2} = \frac{-4}{-6} = \frac{2}{3} \]

Now, we compare:
\[ \frac{7}{5} \neq \frac{2}{3} \]

Since \( \frac{7}{5} \) is not equal to \( \frac{2}{3} \), the condition for a unique solution is satisfied.

### Step 3: Solve the Equations
Since we have established that there is a unique solution, we will solve the equations using the elimination method.

#### Step 3.1: Multiply the Equations
Multiply Equation 1 by 3 and Equation 2 by 2:
- \( 3 \times (7x - 4y) = 3 \times 49 \)
\[ 21x - 12y = 147 \quad \text{(Equation 3)} \]

- \( 2 \times (5x - 6y) = 2 \times 57 \)
\[ 10x - 12y = 114 \quad \text{(Equation 4)} \]

#### Step 3.2: Subtract the Equations
Now, subtract Equation 4 from Equation 3:
\[ (21x - 12y) - (10x - 12y) = 147 - 114 \]
This simplifies to:
\[ 11x = 33 \]

#### Step 3.3: Solve for \( x \)
Dividing both sides by 11:
\[ x = 3 \]

### Step 4: Substitute \( x \) back into one of the original equations
Now, substitute \( x = 3 \) back into Equation 1:
\[ 7(3) - 4y = 49 \]
This simplifies to:
\[ 21 - 4y = 49 \]
Rearranging gives:
\[ -4y = 49 - 21 \]
\[ -4y = 28 \]
Dividing by -4:
\[ y = -7 \]

### Final Solution
Thus, the unique solution to the given pair of linear equations is:
\[ x = 3, \quad y = -7 \]