To find the missing number that makes the equation have infinitely many solutions, we first need to understand that for an equation to have infinitely many solutions, the two sides of the equation must be identical for all values of the variable $x$.

Given the equation: $x + 16 = 2(-x + 8)$

Let's simplify and solve for $x$.

1. Distribute the 2 on the right side: $x + 16 = 2(-x + 8)$ $x + 16 = -2x + 16$

2. Move all terms involving $x$ to one side: $x + 2x = 16 - 16$ $3x = 0$

3. Solve for $x$: $x = 0$

Since solving for $x$ gives us $x = 0$ as a specific solution rather than showing the equation is true for all $x$, this indicates we have made a small assumption about the form. To ensure the equation holds for all $x$, let's revisit the initial question to ensure we correctly interpret it to find a form that does not just solve for $x$ specifically but instead makes both sides identically equal for any $x$.

Looking back: $x + 16 = 2(-x + b)$

We want to find $b$ so the equation is true for all $x$. The expanded equation is: $x + 16 = -2x + 2b$

To make both sides equal for all $x$, the coefficients of $x$ and constant terms must match:

• Coefficient of $x$: $1$ on the left, $-2$ on the right.
• Constant terms: $16$ on the left, $2b$ on the right.

This implies:

$1 = -2 \quad \text{(Not possible)}$

And:

$16 = 2b$

Solving $2b = 16$:

$b = 8$

Thus, the correct missing number $b$ in $2(-x + b)$ should be 8 for the equation to become true for any $x$.

Now, the correct equation is:

$x + 16 = 2(-x + 8)$

Upon distributing:

$x + 16 = -2x + 16$

Rearranging shows it simplifies to $3x = 0$, but for the coefficients to match, notice if $b = 8$ doesn't make it identical for all $x$ due to needing the same coefficients on both $x$, then it confirms we look for coefficients first.

Therefore, both should match:

Thus, when we set $b = 8$ earlier, it aligned to ensure both constant sides match yet $x + 2x = 0$ makes $3x = 0$, not all. The point is infinitely many solutions should match for both. The correct interpretation matches by having no unknowns left just to keep consistent.

Conclusion

The missing number that makes the equation infinitely many solutions cannot exist unless it had kept $1 = -2$ an impossible check directly, thus verifying no value works for both sides identical.

Questions:

1. How would you solve if asked for one specific solution and not infinitely many?
2. Can there exist systems that do align identical both $x$ values too?
3. Would any non-linear be solved similarly for infinite answers?
4. Can matrices represent these equations ever if extended like $ax + b = c(d - e)$?
5. Could we see step-by-step fully expanded terms align directly?

Tip: Always ensure any values multiply back consistently when solving for matching both sides of an equation.