#### Answer

The number of touchdowns is 6, extra-point kicks is 6 and field goals is 1.

#### Work Step by Step

Add the scores of both teams to get the total score$\rightarrow 35+10=45 $
The number of touchdowns, extra-point kicks, and field goals are unknown but we know the ratio between them.
We mark the extra-point kicks as our $x$. There is the same number of touchdowns as extra-point kicks, therefore the number of touchdowns is $x$ as well.
There were six times as many touchdowns
as field goals which means that the number of field goals can be expressed as $\frac{x}{6}$.
Now we have the proportion: $x:x:\frac{x}{6}$
We assign them the points they are worth of and we get $6\times x$ for touchdowns, $1\times x$ for extra-point kicks and $3\times \frac{x}{6}$ for field goals.
We know that $6\times x$, $1\times x$ and $3\times \frac{x}{6}$ should be equal to the total points scored which is 45. From these data we can make an equation:
$6x+x+\frac{3x}{6}=45$
After simplification we get $x=6$
Using the proportion we can calculate that the number of touchdowns is 6, extra-point kicks is 6 and field goals is 1.