5.
Let x be the age of the father and y be the age of the son. As of today, he's 3 times older, so we have [tex]x=3y[/tex]
10 years ago their ages were, respectively, x-10 and y-10, and the father was 5 times older: [tex]x-10 = 5(y-10)[/tex]
So, we have the system
[tex]\begin{cases} x=3y\\x-10=5(y-10)\end{cases}[/tex]
Using the first equation, we can substitute every occurrence of "x" with "3y" in the second equation:
[tex]3y-10=5(y-10) \iff 3y-10=5y-50 \iff 2y = 40 \iff y=20[/tex]
So, the son is 20 years old, which means that the father is 60 years old.
Indeed, 10 years ago they were 10 and 50 years old, so the father was 5 times older.
6.
Let x be the age of the grandfather and y the age of the granddaughter. We know that the grandfather is 10 times older: [tex]x=10y[/tex]
He also is 54 years older: [tex]x=y+54[/tex]
Again, if we substitute x=10y in the second equation we have
[tex]10y=y+54 \iff 9y=54 \iff y=6[/tex]
