For what value of `C_(2)` the system of equation ` 6 x + 2y = 2 & 3 x + y = C_(2)` will be coincident ?
A. 4
B. 0
C. 2
D. 1

Options

A. A
B. B
C. C
D. D

Answer

To determine the value of \( C_2 \) for which the system of equations

1. \( 6x + 2y = 2 \)
2. \( 3x + y = C_2 \)

is coincident, we need to analyze the conditions for two lines to be coincident. Two lines are coincident if their corresponding coefficients are proportional.

### Step 1: Write the equations in standard form
The equations are already in standard form:
- Equation 1: \( 6x + 2y = 2 \)
- Equation 2: \( 3x + y = C_2 \)

### Step 2: Identify the coefficients
From the first equation \( 6x + 2y = 2 \), the coefficients are:
- \( a_1 = 6 \)
- \( b_1 = 2 \)
- \( c_1 = 2 \)

From the second equation \( 3x + y = C_2 \), the coefficients are:
- \( a_2 = 3 \)
- \( b_2 = 1 \)
- \( c_2 = C_2 \)

### Step 3: Set up the proportion
For the lines to be coincident, 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 have:
\[ \frac{6}{3} = \frac{2}{1} = \frac{2}{C_2} \]

### Step 4: Simplify the ratios
Calculating the left side:
\[ \frac{6}{3} = 2 \]
Calculating the second ratio:
\[ \frac{2}{1} = 2 \]

Now we have:
\[ 2 = \frac{2}{C_2} \]

### Step 5: Solve for \( C_2 \)
Cross-multiplying gives:
\[ 2C_2 = 2 \]
Dividing both sides by 2:
\[ C_2 = 1 \]

### Conclusion
Thus, the value of \( C_2 \) for which the system of equations is coincident is \( \boxed{1} \).