Chapter 7.3, Problem 59E

To determine

The number of touchdowns, extra-point kicks, and field goals scored.

Answer to Problem 59E

  $6$ touchdowns, $6$ extra-point kicks and $1$ field goal.

Explanation of Solution

Given information:

In Super Bowl I, on January 15, 1967, theGreen Bay Packers defeated the Kansas City Chiefs by a score of 35 to 10. The total points scored came from a combination of touchdowns, extra-point kicks, and field goals, worth 6, 1, and 3 points, respectively. The numbers of touchdowns and extra-point kicks were equal. There were six times as many touchdowns as field goals.

Calculation:

From the given information, there are three equations with three unknowns:

Let touchdowns $(A)=6$

Let extra-point kicks $(B)=1$

Let field goals $(C)=3$

The equations are:

  $\begin{array}{l}6A+B+3C=\mathrm{45..........}(1)\\ A+B+C=\mathrm{13..............}(2)\\ A=6C\mathrm{..........................}(3)\end{array}$

Subtract equation first by equation second to get the equation fourth:

  $\begin{array}{l}6A-A+B-B+3C-C=45-13\\ 5A+2C=\mathrm{32............}(4)\end{array}$

Substitute equation third into equation forth to solve for $C$ :

  $\begin{array}{l}5(6C)+2C=32\\ 30C+2C=32\\ 32C=32\\ C=1\end{array}$

Substitute $C$ into equation fourth to solve for $A$ :

  $\begin{array}{l}5A+2(1)=32\\ 5A=32-2\\ A=6\end{array}$

Substitute $A$ and $C$ into equation first or equation second to solve for $B$ :

  $\begin{array}{l}6+B+1=13\\ B=6\end{array}$

Thus, there are $6$ touchdowns, $6$ extra-point kicks and $1$ field goal.