The Fabulous Footballers scored an incredible $55$ points at last night’s game. Interestingly, the number of field goals was $1$ more than twice the number of touchdowns. The Fabulous Footballers earned $7$ points for each touchdown and $3$ points for each field goal.  

1. Multiple Choice: Which system of equations below best represents this situation? Explain your reasoning. Assume that $t$ represents the number of touchdowns and $f$ represents the number of field goals.

   1.  $t=2f+1$
      $7t+3f=55$

   2.  $f=2t+1$
      $7t+3f=55$ 

   3.  $t=2f+1$
      $3t+7f=55$ 

   4.  $f=2t+1$
      $3t+7f=55$ 

   Hint (a):

   $7$ times the number of touchdowns and $3$ times the number of field goals equals the total number of points.
   The number of field goals is two times the number of touchdowns plus one.

   Answer (a):

   The only set of equations that meets all these requirements is (ii).

2. Solve the system you selected in part (a) and determine how many touchdowns and field goals the Fabulous Footballers earned last night.

   Hint 1(b):

   Substitute $2t+1$ in for $f$.
   $7t+3(2t+1)=55$

   Hint 2(b):

   Solve for $t$.
   $7t+6t+3=55$
   $13t=52$

   Answer 1(b):

   $t=4$ touchdowns

   Hint 3(b):

   Substitute your answer for $t$ back into the equation for $f$.
   $f=2(4)+1$

   Answer 2(b):

   $f=9$ field goals