If `a * b = a+b+7` for all `a,b in z` then inverse element of 9 is:

Options

A. 23
B. `-9`
C. `-23`
D. 24

Answer

To find the inverse element of 9 under the operation defined by \( a * b = a + b + 7 \), we will follow these steps:

### Step 1: Identify the operation
We are given the operation \( a * b = a + b + 7 \).

### Step 2: Find the identity element
The identity element \( e \) for the operation must satisfy the equation:
\[ a * e = a \]
Substituting the operation into this equation, we have:
\[ a + e + 7 = a \]

### Step 3: Solve for the identity element
To isolate \( e \), we can subtract \( a \) from both sides:
\[ e + 7 = 0 \]
Now, subtract 7 from both sides:
\[ e = -7 \]
Thus, the identity element is \( -7 \).

### Step 4: Define the inverse element
The inverse element \( d \) of an element \( a \) must satisfy:
\[ a * d = e \]
In our case, we need to find the inverse of \( 9 \):
\[ 9 * d = -7 \]

### Step 5: Substitute the operation
Using the operation definition:
\[ 9 + d + 7 = -7 \]

### Step 6: Solve for \( d \)
Combine like terms:
\[ 9 + d + 7 = -7 \]
This simplifies to:
\[ d + 16 = -7 \]
Now, subtract 16 from both sides:
\[ d = -7 - 16 \]
\[ d = -23 \]

### Conclusion
The inverse element of \( 9 \) is \( -23 \).

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