In Super Bowl I, the Green Bay Packers defeated the Kansas City chiefs by a score of 35 to 10. The total points came from 13 different scoring plays, which were a combination of touchdowns, extra point kicks, and field goals, worth 6, 1, and 3 points respectively.The same number of touchdowns and extra point kicks were scored. There were six times as many touchdowns as field goals. How many touchdowns, extra point kicks, and field goals were scored during the game?
We have to solve this using Gaussian Elimination and by using the variables x, y, and z I have determined that one fo the 3 equations is 6x+6y+z=13.
I am stuck after that.
6t + 1x + 3f = 35 + 10 = 45...(A)
t + x + f = 13...(B)
t = x...t - x = 0...(C)
(A) - 3 (B)...3t - 2x = 6
subtracting 2(C)...t = 6
substitute back to find x and f
To solve the system of equations, we can use the information given. Let's assign the variables as follows:
x = number of touchdowns
y = number of extra point kicks
z = number of field goals
From the first sentence in the question, we know that the total score was 35, which means the equation is:
6x + 1y + 3z = 35
We know that the total number of scoring plays was 13, so we have another equation:
x + y + z = 13
Finally, we are given two additional pieces of information:
1. The same number of touchdowns and extra point kicks were scored:
x = y
2. There were six times as many touchdowns as field goals:
x = 6z
Now, we can rewrite the system of equations:
6x + y + 3z = 35 (1)
x + y + z = 13 (2)
x - y = 0 (3)
x - 6z = 0 (4)
To solve the system using Gaussian elimination, we can start by eliminating variables. Let's eliminate y using equations (2) and (3):
(2) - (3): (x + y + z) - (x - y) = 13 - 0
2y + z = 13
Now, let's eliminate x using equations (3) and (4):
(3) + (4): (x - y) + (x - 6z) = 0 + 0
2x - 6z = 0
Now, we can substitute the value of x from equation (4) into equation (2):
6z - y + z = 35
7z - y = 35
From equation (4), we know that x - 6z = 0, so x = 6z. Substituting this into equation (2), we get:
6z = y
Now we can substitute the values of y and x into equation (3):
x - y = 0
6z - 6z = 0
0 = 0
The equation 0 = 0 indicates that z can take any value. Let's assign a value to z and solve for x and y.
Assume z = 1:
6(1) = x
6 = x
6 = y
Therefore, for z = 1, we have x = 6, y = 6.
So the solution is:
Number of touchdowns (x) = 6
Number of extra point kicks (y) = 6
Number of field goals (z) = 1
Thus, during the game, 6 touchdowns, 6 extra point kicks, and 1 field goal were scored.