Final answer:
Juan is 20 years old and Emily is 14 years old.
Step-by-step explanation:
To solve this problem, we can set up a system of equations.
Let's represent Emily's age as 'E' and Juan's age as 'J'.
According to the problem, Emily is six years younger than Juan. So we can write the equation: E = J - 6.
We are also given that the sum of their ages is 34, so J + E = 34.
Now we can solve this system of equations using substitution, elimination, or any other preferred method.
Let's substitute the value of E from the first equation into the second equation: J - 6 + J = 34.
Combining like terms, we get: 2J - 6 = 34.
Next, we can isolate J by adding 6 to both sides: 2J = 40.
Dividing both sides by 2, we find that J = 20.
Finally, we can substitute the value of J back into the first equation to find E: E = 20 - 6 = 14.
Therefore, Juan is 20 years old and Emily is 14 years old.