For the real values of c , the pair of equations
x - 2y = 8
5x - 10 y = c
have a unique solution . Justify whether it is true or false.

Answer

To determine whether the given pair of equations has a unique solution for real values of \( c \), we will analyze the equations step by step.

### Step 1: Write the equations
The given equations are:
1. \( x - 2y = 8 \)
2. \( 5x - 10y = c \)

### Step 2: Rewrite the equations in standard form
We can rewrite both equations in the standard form \( Ax + By + C = 0 \).

1. For the first equation:
\[ x - 2y - 8 = 0 \quad \Rightarrow \quad 1x - 2y - 8 = 0 \]
Here, \( A_1 = 1, B_1 = -2, C_1 = -8 \).

2. For the second equation:
\[ 5x - 10y - c = 0 \quad \Rightarrow \quad 5x - 10y - c = 0 \]
Here, \( A_2 = 5, B_2 = -10, C_2 = -c \).

### Step 3: Determine the condition for a unique solution
For a pair of linear equations to have a unique solution, the following condition must be satisfied:
\[ \frac{A_1}{A_2} \neq \frac{B_1}{B_2} \]
This means that the ratios of the coefficients of \( x \) and \( y \) must not be equal.

### Step 4: Calculate the ratios
Now we calculate the ratios:
1. \( \frac{A_1}{A_2} = \frac{1}{5} \)
2. \( \frac{B_1}{B_2} = \frac{-2}{-10} = \frac{2}{10} = \frac{1}{5} \)

### Step 5: Analyze the results
Since:
\[ \frac{A_1}{A_2} = \frac{1}{5} \quad \text{and} \quad \frac{B_1}{B_2} = \frac{1}{5} \]
we find that:
\[ \frac{A_1}{A_2} = \frac{B_1}{B_2} \]
This means that the two equations are not independent; they are proportional.

### Step 6: Conclusion
Since the ratios of the coefficients of \( x \) and \( y \) are equal, the equations represent parallel lines, which means they do not intersect. Therefore, there is no unique solution for any real value of \( c \).

### Final Answer
The statement that for the real values of \( c \), the pair of equations has a unique solution is **false**.

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