This season, Lisa's lacrosse team has won $\frac 23$ of their home games (games played at Lisa's school), but just $\frac 25$ of their away games (games played at other schools). In total, Lisa's team has won $26$ games out of $49$ games they have played. How many home games has Lisa's team played? Explain how you solved the problem.

The team has played 24 home games.

Let h be the number of home games and a be the number of away games.  The total number of games is 49; this gives us
h+a=49

They won 2/3 of the home games and 2/5 of the away games; there were 26 games won.  This gives us
2/3h + 2/5a = 26

In the first equation we isolate h by subtracting a from both sides:
h+a-a=49-a
h=49-a

Substitute this into the second equation:
2/3(49-a)+2/5a = 26

Using the distributive property, we have
2/3*49 - 2/3*a + 2/5a = 26
98/3 - 2/3a + 2/5a = 26

Finding a common denominator to combine like terms, we have
98/3 - 10/15a + 6/15a = 26
98/3 - 4/15a = 26

We want to convert the whole number to thirds as well; 26 = 26*3/3 = 78/3:
98/3 - 4/15a = 78/3

Subtracting 98/3 from both sides:
98/3 - 4/15a - 98/3 = 78/3 - 98/3
-4/15a = -20/3

Divide both sides by -4/15:
a = -20/3 ÷ -4/15
a = -20/3 × - 15/4 = 300/12 = 25

There were 25 away games.

This means there were 49-25 = 24 home games.