Question

Recall that in football, touchdowns are worth 7 points each (including the extra point), and field goals are worth 3 points each. List all of the different ways a football team could score 63 points, if they make at least one touchdown and at least one field goal.

Answer 1

63 points

1) 3 touchdowns and 14 field goals

2) 6 touchdowns and 7 field goals

Answer 2

The different ways a football team cold score 63 points are by having;

1) 3 touchdowns and 14 field goals

2) 6 touchdowns and 7 field goals

Explanation:

The given parameters are;

The worth of each touchdown = 7 point

The worth of each field goal = 3 points

The number of touch down ≥ 1

The number of field goals ≥ 1

Let x represent the number of touchdowns and let y represent the number of field goals, we have;

7·x + 3·y = 63...(1)

x + y ≥ 2...(1)

Making y the subject of the equation, 7·x + 3·y = 63, gives;

3·y = 63 - 7·x

y = 21 - (7/3)·x

Making y the subject of the inequality, x + y ≥ 2, gives;

y ≥ 2 - x

From the attached graph created with Microsoft Excel, we have the points where x ≥ 1, that have a whole number value of y ≥ 0 as follows;

1) When x = 3, y = 14 gives 7×3 + 3×14 = 63

2) When x = 6, y = 7 gives 7×6 + 3×7 = 63

Therefore, the different ways a football team cold score 63 points are;

1) 3 touchdowns and 14 field goals

2) 6 touchdowns and 7 field goals.