Find c if the system of equations cx + 3y + (3 - c) = 0 , 12 x + cy - c = 0 has infinitely many solutions ?

Answer

To find the value of \( c \) such that the system of equations

1. \( cx + 3y + (3 - c) = 0 \)
2. \( 12x + cy - c = 0 \)

has infinitely many solutions, we need to ensure that the coefficients of the equations are proportional.

### Step-by-step Solution:

**Step 1: Write the equations in standard form.**

The equations can be rewritten as:
1. \( cx + 3y = c - 3 \)
2. \( 12x + cy = c \)

**Hint:** Rearranging the equations helps in identifying the coefficients clearly.

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**Step 2: Identify the coefficients.**

From the first equation \( cx + 3y = c - 3 \), the coefficients are:
- Coefficient of \( x \): \( c \)
- Coefficient of \( y \): \( 3 \)
- Constant term: \( c - 3 \)

From the second equation \( 12x + cy = c \), the coefficients are:
- Coefficient of \( x \): \( 12 \)
- Coefficient of \( y \): \( c \)
- Constant term: \( c \)

**Hint:** Coefficients are essential for establishing the proportionality condition.

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**Step 3: Set up the proportionality condition.**

For the equations to have infinitely many solutions, the ratios of the coefficients must be equal:
\[ \frac{c}{12} = \frac{3}{c} = \frac{c - 3}{-c} \]

**Hint:** Remember that for two lines to coincide, the ratios of their coefficients must be equal.

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**Step 4: Solve the first ratio \( \frac{c}{12} = \frac{3}{c} \).**

Cross-multiplying gives:
\[ c^2 = 36 \]
Thus,
\[ c = 6 \quad \text{or} \quad c = -6 \]

**Hint:** When cross-multiplying, ensure to check both positive and negative roots.

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**Step 5: Solve the second ratio \( \frac{3}{c} = \frac{c - 3}{-c} \).**

Cross-multiplying gives:
\[ 3(-c) = c(c - 3) \]
This simplifies to:
\[ -3c = c^2 - 3c \]
Rearranging leads to:
\[ 0 = c^2 \]
Thus,
\[ c = 0 \]

**Hint:** Always simplify equations carefully to avoid losing solutions.

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**Step 6: Analyze the results.**

From the first ratio, we found \( c = 6 \) or \( c = -6 \). From the second ratio, we found \( c = 0 \).

To have infinitely many solutions, we need to check which of these values satisfy both conditions.

**Final Decision:**

The only consistent value that satisfies the proportionality condition for both equations is \( c = 6 \).

### Final Answer:
Thus, the value of \( c \) for which the system of equations has infinitely many solutions is \( c = 6 \).

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