Answer
a) $3$ hours
b) The costs will never be the same.
Work Step by Step
a) Let's note by $x$ the number of painting hours and by $y$ the total cost of a piece. We have:
Studio $A$: $$y = 8x + 10\tag{1}$$ Studio $B$: $$y = 6x + 16\tag{2}$$ We determine $x$ so that the cost is the same in both studios by equating $(1)$ and $(2)$:
$$\begin{align}
8x + 10& = 6x + 16\\
2x + 10& = 16\\
2x& = 6\\
x& = 3.
\end{align}$$ Therefore, after $3$ hours of painting, the total costs are the same at both studios.
b) In this case we have:
Studio $A$: $$y = 8x + 10\tag{1}$$ Studio $B$: $$y = 8x + 16\tag{2}$$ We determine $x$ so that the cost is the same in both studios by equating $(1)$ and $(2)$:
$$\begin{align}
8x + 10& = 8x + 16\\
10& = 16\\
0&=6.
\end{align}$$ We reached an untrue statement. Therefore, this equation has no solution.
Here is what is happening in context: If Studio $B$ increases the hourly studio fee by $2$ dollars, then both Studio $A$ and Studio $B$ will charge the same hourly rate of $8$ dollars. But the vase still costs $6$ dollars more at Studio $B$ than at Studio $A$, and that number never changes. So, it will always be more expensive to paint at Studio $B$ than at Studio $A$, and therefore it is impossible for painting to cost the same at both studios after any number of hours.
