If a*b means a is added to b,
a $ b means b is subtracted from a.
a # b means a is multiplied by b.
a & b means a is divided by b,
then answer the following questions :
If `a= 4x-7xy+3y+12" and "b= 12x-9xy+6y-3`, then find a $ b # 3.

Options

A. `-32x+20xy-15y+21`
B. `32x+4xy-y-12`
C. `-32x-xy+13`
D. `-3xy+x+12`

Answer

To solve the problem step by step, we will first define the operations as per the question and then substitute the values of \( a \) and \( b \) accordingly.

### Given:
- \( a = 4x - 7xy + 3y + 12 \)
- \( b = 12x - 9xy + 6y - 3 \)

### Operations:
- \( a * b \) means \( a + b \)
- \( a \$ b \) means \( a - b \)
- \( a \# b \) means \( a \times b \)
- \( a \& b \) means \( a \div b \)

### We need to find:
\[ a \$ (b \# 3) \]

### Step 1: Calculate \( b \# 3 \)
Using the definition of \( \# \):
\[ b \# 3 = b \times 3 \]
Substituting the value of \( b \):
\[ b \# 3 = (12x - 9xy + 6y - 3) \times 3 \]

### Step 2: Distribute \( 3 \) to each term in \( b \)
\[ b \# 3 = 3(12x) - 3(9xy) + 3(6y) - 3(3) \]
\[ = 36x - 27xy + 18y - 9 \]

### Step 3: Now calculate \( a \$ (b \# 3) \)
Using the definition of \( \$ \):
\[ a \$ (b \# 3) = a - (b \# 3) \]
Substituting the value of \( a \) and \( b \# 3 \):
\[ a \$ (b \# 3) = (4x - 7xy + 3y + 12) - (36x - 27xy + 18y - 9) \]

### Step 4: Distribute the negative sign
\[ = 4x - 7xy + 3y + 12 - 36x + 27xy - 18y + 9 \]

### Step 5: Combine like terms
- For \( x \) terms: \( 4x - 36x = -32x \)
- For \( xy \) terms: \( -7xy + 27xy = 20xy \)
- For \( y \) terms: \( 3y - 18y = -15y \)
- For constant terms: \( 12 + 9 = 21 \)

### Final Expression:
Combining all the results gives:
\[ a \$ (b \# 3) = -32x + 20xy - 15y + 21 \]

### Conclusion:
The final answer is:
\[ -32x + 20xy - 15y + 21 \]

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