Calculate the value of c for which pair of linear equations cx - y = 2 and 6x - 2y = 4 will have infinitely many solutions.

Options

A. 3
B. 5
C. -1
D. 0

Answer

To find the value of \( c \) for which the pair of linear equations \( cx - y = 2 \) and \( 6x - 2y = 4 \) will have infinitely many solutions, we need to determine when the two equations represent the same line. This occurs when the equations are proportional.

### Step 1: Rewrite the equations in standard form
The given equations are:
1. \( cx - y = 2 \)
2. \( 6x - 2y = 4 \)

### Step 2: Rearranging the second equation
We can simplify the second equation:
\[ 6x - 2y = 4 \implies 3x - y = 2 \quad \text{(by dividing the entire equation by 2)} \]

### Step 3: Set the equations equal for proportionality
Now we have:
1. \( cx - y = 2 \)
2. \( 3x - y = 2 \)

For the two equations to be equivalent (represent the same line), the coefficients of \( x \) and \( y \) must be proportional:
\[ \frac{c}{3} = \frac{-1}{-1} = 1 \]

### Step 4: Solve for \( c \)
From the proportionality condition:
\[ \frac{c}{3} = 1 \]
Multiplying both sides by 3 gives:
\[ c = 3 \]

### Conclusion
Thus, the value of \( c \) for which the pair of linear equations will have infinitely many solutions is:
\[ \boxed{3} \]