Do the following pair of linear equations have no solution? Justify your answer.
`x= 2y, y= 2x`

Answer

To determine whether the given pair of linear equations has no solution, we will analyze the equations step by step.

**Given equations:**
1. \( x = 2y \)
2. \( y = 2x \)

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

The standard form of a linear equation is \( Ax + By + C = 0 \).

- For the first equation \( x = 2y \):
\[ x - 2y = 0 \]
This can be written as \( 1x - 2y + 0 = 0 \), where \( A_1 = 1, B_1 = -2, C_1 = 0 \).

- For the second equation \( y = 2x \):
\[ y - 2x = 0 \]
This can be written as \( -2x + 1y + 0 = 0 \), where \( A_2 = -2, B_2 = 1, C_2 = 0 \).

**Step 2: Identify the coefficients.**

From the standard forms, we have:
- \( A_1 = 1, B_1 = -2, C_1 = 0 \)
- \( A_2 = -2, B_2 = 1, C_2 = 0 \)

**Step 3: Check the condition for no solution.**

For a pair of linear equations to have no solution, the following condition must hold:
\[ \frac{A_1}{A_2} = \frac{B_1}{B_2} \neq \frac{C_1}{C_2} \]

Calculating \( \frac{A_1}{A_2} \) and \( \frac{B_1}{B_2} \):
\[ \frac{A_1}{A_2} = \frac{1}{-2} = -\frac{1}{2} \]
\[ \frac{B_1}{B_2} = \frac{-2}{1} = -2 \]

Now, we compare:
- \( -\frac{1}{2} \neq -2 \) (True)

Next, we check \( \frac{C_1}{C_2} \):
\[ \frac{C_1}{C_2} = \frac{0}{0} \]
This is undefined.

**Step 4: Conclusion.**

Since \( \frac{A_1}{A_2} \neq \frac{B_1}{B_2} \), we conclude that the pair of linear equations does not satisfy the condition for no solution. Therefore, the given pair of equations has a unique solution.

**Final Answer:**
The given pair of linear equations does not have no solution; instead, it has a unique solution.

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