To find the number of visits (denoted as \( x \)) at which the costs of the two plans are equal, we can set up the following equations based on the given information:
- Cost for the first option (pay per visit): \( 10x \)
- Cost for the second option (monthly fee plus per visit): \( 32 + 6x \)
Now we set these two costs equal to each other:
\[ 10x = 32 + 6x \]
Next, we'll solve this equation. First, we can subtract \( 6x \) from both sides:
\[ 10x - 6x = 32 \] \[ 4x = 32 \]
Now, divide both sides by 4:
\[ x = \frac{32}{4} = 8 \]
So, it would take 8 visits for the cost of the two plans to be equal.
The correct equation from the options provided is: \[ 10x = 32 + 6x \quad ; \quad x = 8 \]