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 statement "If \( x = 6 \) and \( y = -2 \), then \( x - 2y = 9 \)", we will follow these steps:

### Step 1: Identify the Statements
Let:
- \( p \): \( x = 6 \)
- \( q \): \( y = -2 \)
- \( r \): \( x - 2y = 9 \)

### Step 2: Write the Original Statement
The original statement can be expressed as:
\[ p \land q \implies r \]
This means "If \( p \) and \( q \) are true, then \( r \) is true."

### Step 3: Apply the Contrapositive Rule
The contrapositive of a statement \( a \implies b \) is given by \( \neg b \implies \neg a \). In our case, we need to negate both \( r \) and \( p \land q \).

### Step 4: Negate the Conclusion
Negating \( r \) gives us:
\[ \neg r: x - 2y \neq 9 \]

### Step 5: Negate the Premise
Negating \( p \land q \) gives us:
\[ \neg (p \land q) = \neg p \lor \neg q \]
Where:
- \( \neg p: x \neq 6 \)
- \( \neg q: y \neq -2 \)

Thus, we have:
\[ \neg (p \land q) = x \neq 6 \lor y \neq -2 \]

### Step 6: Combine the Negations
Now we can combine our results:
\[ \neg r \implies \neg (p \land q) \]
This translates to:
\[ x - 2y \neq 9 \implies (x \neq 6 \lor y \neq -2) \]

### Step 7: Write the Final Contrapositive Statement
The contrapositive statement in words is:
"If \( x - 2y \neq 9 \), then \( x \neq 6 \) or \( y \neq -2 \)."

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

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