If x = 6 and `y=-2` then `x-2y=9.` The contrapositive of this statement is

Options

A. If `x-2y ne 9` then `xne 6 or y ne -2`
B. If `x-2y ne 9` then `x ne 6 and y ne -2`
C. If `x-2y =9` then `x=6 and y=-2`
D. None of these

Answer

To find the contrapositive of the given statement "If \( x = 6 \) and \( y = -2 \), then \( x - 2y = 9 \)", we can follow these steps:

### Step 1: Identify the Statements
Let's denote:
- \( P \): \( x = 6 \)
- \( Q \): \( y = -2 \)
- \( R \): \( x - 2y = 9 \)

The original statement can be expressed as:
\[ P \land Q \implies R \]

### Step 2: Write the Negations
Next, we need to find the negations of these statements:
- Negation of \( R \) (i.e., \( x - 2y \neq 9 \))
- Negation of \( P \) (i.e., \( x \neq 6 \))
- Negation of \( Q \) (i.e., \( y \neq -2 \))

### Step 3: Formulate the Contrapositive
The contrapositive of a statement of the form \( A \implies B \) is \( \neg B \implies \neg A \). In our case, we need to negate the conclusion \( R \) and the premise \( P \land Q \).

Thus, the contrapositive will be:
\[ \neg R \implies \neg (P \land Q) \]

Using De Morgan's laws, we can express \( \neg (P \land Q) \) as:
\[ \neg P \lor \neg Q \]

So, the contrapositive becomes:
\[ x - 2y \neq 9 \implies (x \neq 6 \lor y \neq -2) \]

### Final Statement
The contrapositive of the original statement is:
"If \( x - 2y \neq 9 \), then \( x \neq 6 \) or \( y \neq -2 \)."

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