For some real numbers $$ a $$ and $$ b $$, the system of equations
$$
x + y = 4
$$
and
$$
(a + 5)x + (b^2 - 15)y = 8b
$$
has infinitely many solutions for $$ x $$ and $$ y $$. Then, the maximum possible value of $$ ab $$ is:

Question

For some real numbers $$ a $$ and $$ b $$, the system of equations  
$$
x + y = 4
$$
and  
$$
(a + 5)x + (b^2 - 15)y = 8b
$$  
has infinitely many solutions for $$ x $$ and $$ y $$. Then, the maximum possible value of $$ ab $$ is:

Accepted Answer

To find the maximum possible value of $$ ab $$, we analyze the given system of equations:

1. **Conditions for infinite solutions**:  
   For the system to have infinitely many solutions, the two equations must be proportional. This implies:  
   $$
   \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}.
   $$
   Comparing coefficients, we get:  
   $$
   \frac{1}{a + 5} = \frac{1}{b^2 - 15} = \frac{4}{8b}.
   $$

2. **Equating proportions**:  
   From $$ \frac{1}{a + 5} = \frac{4}{8b} $$, simplify to:  
   $$
   a + 5 = 2b.
   $$

3. **Substitute into second proportion**:  
   From $$ \frac{1}{a + 5} = \frac{1}{b^2 - 15} $$, equate:  
   $$
   a + 5 = b^2 - 15.
   $$
   Substitute $$ a + 5 = 2b $$ into this equation:  
   $$
   2b = b^2 - 15.
   $$

4. **Solve quadratic equation**:  
   Rearrange to form:  
   $$
   b^2 - 2b - 15 = 0.
   $$
   Factorize:  
   $$
   (b - 5)(b + 3) = 0.
   $$
   Thus, $$ b = 5 $$ or $$ b = -3 $$.

5. **Find corresponding values of $$ a $$**:  
   - If $$ b = 5 $$:  
     Substitute into $$ a + 5 = 2b $$:  
     $$ a + 5 = 10 $$, so $$ a = 5 $$.  
     Then, $$ ab = 5 	imes 5 = 25. $$
   - If $$ b = -3 $$:  
     Substitute into $$ a + 5 = 2b $$:  
     $$ a + 5 = -6 $$, so $$ a = -11 $$.  
     Then, $$ ab = -11 	imes -3 = 33. $$

6. **Conclusion**:  
The maximum possible value of $$ ab $$ is **33**.

Answer

Answer

Answer