Form the pair of linear equations for the following problems and find their solution by substitution method.
A fraction becomes `(9)/(11)`, if 2 is added to both the numerator and the denominator. If, 3 is added to both the numerator and the denominator it becomes `(5)/(6)`. Find the fraction.

Answer

To solve the problem, we need to form a pair of linear equations based on the given conditions and then solve them using the substitution method. Let's break down the steps:

### Step 1: Define Variables
Let:
- \( x \) = numerator of the fraction
- \( y \) = denominator of the fraction

### Step 2: Form the First Equation
According to the problem, if 2 is added to both the numerator and the denominator, the fraction becomes \( \frac{9}{11} \). This can be expressed as:
\[ \frac{x + 2}{y + 2} = \frac{9}{11} \]
Cross-multiplying gives:
\[ 11(x + 2) = 9(y + 2) \]
Expanding this:
\[ 11x + 22 = 9y + 18 \]
Rearranging the equation:
\[ 11x - 9y + 4 = 0 \quad \text{(Equation 1)} \]

### Step 3: Form the Second Equation
Next, if 3 is added to both the numerator and the denominator, the fraction becomes \( \frac{5}{6} \). This can be expressed as:
\[ \frac{x + 3}{y + 3} = \frac{5}{6} \]
Cross-multiplying gives:
\[ 6(x + 3) = 5(y + 3) \]
Expanding this:
\[ 6x + 18 = 5y + 15 \]
Rearranging the equation:
\[ 6x - 5y + 3 = 0 \quad \text{(Equation 2)} \]

### Step 4: Solve the Equations Using Substitution Method
From Equation 1:
\[ 11x - 9y = -4 \quad \text{(1)} \]
From Equation 2:
\[ 6x - 5y = -3 \quad \text{(2)} \]

We can express \( y \) in terms of \( x \) from Equation 1:
\[ 9y = 11x + 4 \implies y = \frac{11x + 4}{9} \quad \text{(3)} \]

### Step 5: Substitute \( y \) in Equation 2
Substituting Equation (3) into Equation (2):
\[ 6x - 5\left(\frac{11x + 4}{9}\right) = -3 \]
Multiplying through by 9 to eliminate the fraction:
\[ 54x - 5(11x + 4) = -27 \]
Expanding:
\[ 54x - 55x - 20 = -27 \]
Combining like terms:
\[ -x - 20 = -27 \]
Adding 20 to both sides:
\[ -x = -7 \implies x = 7 \]

### Step 6: Find \( y \)
Substituting \( x = 7 \) back into Equation (3):
\[ y = \frac{11(7) + 4}{9} = \frac{77 + 4}{9} = \frac{81}{9} = 9 \]

### Step 7: Conclusion
The fraction is:
\[ \frac{x}{y} = \frac{7}{9} \]

### Final Answer
The required fraction is \( \frac{7}{9} \).

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