The value of c for which the pair of equations `cx -y=2 and 6x- 2y=3` will have infinitely many solutions is

Options

A. 3
B. `-3`
C. `-12`
D. no value

Answer

To find the value of \( c \) for which the pair of equations \( cx - y = 2 \) and \( 6x - 2y = 3 \) will have infinitely many solutions, we can follow these steps:

### Step 1: Write the equations in standard form
The given equations are:
1. \( cx - y = 2 \) (Equation 1)
2. \( 6x - 2y = 3 \) (Equation 2)

We can rewrite these equations in the standard form \( Ax + By + C = 0 \):
1. \( cx - y - 2 = 0 \) (Equation 1)
2. \( 6x - 2y - 3 = 0 \) (Equation 2)

### Step 2: Identify coefficients
From the standard form, we can identify the coefficients:
- For Equation 1: \( A_1 = c, B_1 = -1, C_1 = -2 \)
- For Equation 2: \( A_2 = 6, B_2 = -2, C_2 = -3 \)

### Step 3: Set up the condition for infinitely many solutions
For the system of equations to have infinitely many solutions, the ratios of the coefficients must be equal:
\[ \frac{A_1}{A_2} = \frac{B_1}{B_2} = \frac{C_1}{C_2} \]
Substituting the coefficients we identified:
\[ \frac{c}{6} = \frac{-1}{-2} = \frac{-2}{-3} \]

### Step 4: Simplify the ratios
First, simplify the second ratio:
\[ \frac{-1}{-2} = \frac{1}{2} \]
Now we have:
\[ \frac{c}{6} = \frac{1}{2} \]

### Step 5: Cross-multiply to solve for \( c \)
Cross-multiply to find \( c \):
\[ 2c = 6 \]
Now, divide both sides by 2:
\[ c = 3 \]

### Conclusion
The value of \( c \) for which the pair of equations will have infinitely many solutions is \( \boxed{3} \).

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