To find how many solutions exist for the equation
\[ 3\left(\frac{x}{2} + 5\right) - 6 = \frac{9x + 18}{3}, \]
we will start by simplifying both sides of the equation step by step.
First, simplify the left-hand side:
- Distribute the \(3\) on the left side: \[ 3\left(\frac{x}{2}\right) + 3 \times 5 - 6 = \frac{3x}{2} + 15 - 6 = \frac{3x}{2} + 9. \]
Next, simplify the right-hand side:
- Simplify the right side: \[ \frac{9x + 18}{3} = \frac{9x}{3} + \frac{18}{3} = 3x + 6. \]
Now, rewrite the equation with both sides simplified:
\[ \frac{3x}{2} + 9 = 3x + 6. \]
Next, we will eliminate the fraction by multiplying every term by \(2\) (the denominator):
\[ 2\left(\frac{3x}{2}\right) + 2(9) = 2(3x) + 2(6), \] which simplifies to:
\[ 3x + 18 = 6x + 12. \]
Next, we need to isolate \(x\). Start by moving \(3x\) to the right side:
\[ 18 = 6x - 3x + 12, \]
which reduces to:
\[ 18 = 3x + 12. \]
Now, subtract \(12\) from both sides:
\[ 18 - 12 = 3x, \] yielding:
\[ 6 = 3x. \]
Now, divide both sides by \(3\):
\[ x = 2. \]
Thus, we have found a single solution for the linear equation:
\[ \boxed{1}. \]