If x = 5 and y = -2 , then x -2y =9, the contrapositive of this proposition is

Options

A. If x -2y is not equal to 9 , then ` x ne 5 or y ne -2`
B. ` if x -2y =9, x ne 5 and y ne -2`
C. ` x -2y =9 if and " only " if x=5 and y =-2 `
D. none of these

Explanation

To find the contrapositive of the given proposition, we will follow these steps:

### Step 1: Identify the Statements
The original proposition is:
"If \( x = 5 \) and \( y = -2 \), then \( x - 2y = 9 \)."

We can define:
- Statement \( p \): \( x = 5 \) and \( y = -2 \)
- Statement \( q \): \( x - 2y = 9 \)

### Step 2: Write the Implication
The implication can be written as:
\( p \implies q \)

### Step 3: Form the Contrapositive
The contrapositive of an implication \( p \implies q \) is given by:
\( \neg q \implies \neg p \)

Where:
- \( \neg q \): Negation of \( q \)
- \( \neg p \): Negation of \( p \)

### Step 4: Determine the Negations
1. **Negation of \( q \)**:
- \( q \): \( x - 2y = 9 \)
- \( \neg q \): \( x - 2y \neq 9 \)

2. **Negation of \( p \)**:
- \( p \): \( x = 5 \) and \( y = -2 \)
- \( \neg p \): \( x \neq 5 \) or \( y \neq -2 \)

### Step 5: Write the Contrapositive Statement
Putting it all together, the contrapositive is:
"If \( x - 2y \neq 9 \), then \( x \neq 5 \) or \( y \neq -2 \)."

### Final Answer
The contrapositive of the proposition is:
"If \( x - 2y \neq 9 \), then \( x \neq 5 \) or \( y \neq -2 \)."

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