Find the value of 'p' for which the following pair of linear equations have infinitely many solutions ?

`(p-3)x+3y=p`
`px+py=12`

Options

A. -6
B. 0
C. 6
D. 12

Answer

To find the value of 'p' for which the given pair of linear equations has infinitely many solutions, we need to analyze the equations:

1. \((p-3)x + 3y = p\)
2. \(px + py = 12\)

### Step 1: Rewrite the equations in standard form
We can rewrite the second equation to match the standard form of linear equations:

\[ px + py - 12 = 0 \]

Now we have:

1. \((p-3)x + 3y - p = 0\) (Equation 1)
2. \(px + py - 12 = 0\) (Equation 2)

### Step 2: Identify coefficients
From the equations, we can identify the coefficients:

- For Equation 1:
- \(a_1 = p - 3\)
- \(b_1 = 3\)
- \(c_1 = -p\)

- For Equation 2:
- \(a_2 = p\)
- \(b_2 = p\)
- \(c_2 = -12\)

### Step 3: Set up the condition for infinitely many solutions
For the two equations to have infinitely many solutions, the following condition must hold:

\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \]

Substituting the coefficients, we have:

\[ \frac{p-3}{p} = \frac{3}{p} = \frac{-p}{-12} \]

### Step 4: Solve the first equality
Starting with the first part of the equality:

\[ \frac{p-3}{p} = \frac{3}{p} \]

Cross-multiplying gives:

\[ (p - 3) \cdot p = 3 \cdot p \]

This simplifies to:

\[ p^2 - 3p = 3p \]

Combining like terms:

\[ p^2 - 6p = 0 \]

Factoring out \(p\):

\[ p(p - 6) = 0 \]

This gives us two possible solutions:

\[ p = 0 \quad \text{or} \quad p = 6 \]

### Step 5: Solve the second equality
Now we check the second part of the equality:

\[ \frac{3}{p} = \frac{p}{12} \]

Cross-multiplying gives:

\[ 3 \cdot 12 = p \cdot p \]

This simplifies to:

\[ 36 = p^2 \]

Taking the square root of both sides:

\[ p = 6 \quad \text{or} \quad p = -6 \]

### Step 6: Find common solutions
Now we have the possible values of \(p\):

1. From the first equality: \(p = 0\) or \(p = 6\)
2. From the second equality: \(p = 6\) or \(p = -6\)

The common solution from both sets is:

\[ p = 6 \]

### Conclusion
Thus, the value of \(p\) for which the pair of linear equations has infinitely many solutions is:

\[ \boxed{6} \]