Answer:
24 home games
Explanation:
We can introduce a variable, say, $h$, for the number of home games Lisa's team has played. Since the total number of games they have played is $49$, they must have played $49-h$ away games.
Thus, Lisa's team has won $\frac{2}{3}h$ home games and $\frac{2}{5}(49-h)$ away games. We know they have won a total of $26$ games, so
$$\frac{2}{3}h + \frac{2}{5}(49-h) = 26.$$We can clear fractions from the problem by multiplying both sides of the equation by $3\cdot 5$:
$$\frac{2\cdot 3\cdot 5}{3}h + \frac{2\cdot 3\cdot 5}{5}(49-h) = 26\cdot 3\cdot 5.$$Simplifying this equation, we have
$$10h + 6\cdot (49-h) = 390.$$Expanding the left side using the distributive property, we have
$$10h + 294 - 6h = 390.$$Subtracting $294$ from both sides and simplifying $10h-6h$ to $4h$, we have
$$4h = 96.$$Finally, dividing both sides by $4$, we have $h=24$. Therefore, Lisa's team has played $\boxed{24}$ home games.
We can check that this answer is correct. If Lisa's team has played $24$ home games, then they have won $\frac 23\cdot 24 = 16$ of those games; also, they have played $49-24=25$ away games, and they have won $\frac 25\cdot 25 = 10$ of those games. So, they have won $16+10=26$ games in all, which is what we were expecting.