The contrapositive of `2x + 3y = 9 rArr x != 4` is

Options

A. `x = 4 rArr 2x + 3y != 9`
B. `x = 4 rArr 2x + 3y = 9`
C. `x != 4 rArr 2x + 3y != 9`
D. `x != 4 rArr 2x + 3y = 9`

Answer

To find the contrapositive of the statement "2x + 3y = 9 implies x ≠ 4", we will follow these steps:

### Step 1: Identify the components of the implication
We start with the implication statement:
- Let \( p \) be the statement \( 2x + 3y = 9 \).
- Let \( q \) be the statement \( x \neq 4 \).

### Step 2: Write the implication in logical form
The implication can be written as:
\[ p \implies q \]
This means "if \( p \) is true, then \( q \) is also true".

### Step 3: Determine the contrapositive
The contrapositive of an implication \( p \implies q \) is given by:
\[ \neg q \implies \neg p \]
Where \( \neg q \) is the negation of \( q \) and \( \neg p \) is the negation of \( p \).

### Step 4: Negate the statements
Now we will negate both statements:
- The negation of \( q \) (which is \( x \neq 4 \)) is \( x = 4 \).
- The negation of \( p \) (which is \( 2x + 3y = 9 \)) is \( 2x + 3y \neq 9 \).

### Step 5: Write the contrapositive
Now we can write the contrapositive using the negated statements:
\[ x = 4 \implies 2x + 3y \neq 9 \]

### Final Answer
Thus, the contrapositive of the statement "2x + 3y = 9 implies x ≠ 4" is:
\[ x = 4 \implies 2x + 3y \neq 9 \]

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