Answers:
6 touchdowns (TD)
3 field goals (FG)
5 points after touchdown (PAT)
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Step-by-step explanation:
Acronyms used in football
- PAT = point after touchdown
- TD = touchdown
- FG = field goal
Define the following variables
- x = number of TDs
- y = number of FGs
- z = number of PATs
"They had 3 more touchdowns than field goals", so,
TD = FG + 3
x = y + 3
They scored 14 times which means we can say
x+y+z = 14
Let's look at the point breakdown per type of score
- TDs give 6 points each. Scoring x of them gives 6x points
- FGs give 3 points each. Scoring y of them gives 3y points
- PATs give 1 point each. Scoring z of them gives 1z points
Then we can add up those expressions to get us to 6x+3y+1z = 50 points total. This is the same as saying 6x+3y+z = 50.
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Here's our system of equations
x = y+3
x+y+z = 14
6x+3y+z = 50
Let's replace each x with y+3 in the second equation. Then solve for z.
x+y+z = 14
y+3+y+z = 14
2y+z = 14-3
2y+z = 11
z = 11-2y
Now replace every x with y+3 in the third original equation
6x+3y+z = 50
6(y+3)+3y+z = 50
6y+18+3y+z = 50
9y+z = 50-18
9y+z = 32
Then plug in z = 11-2y and solve for y
9y+z = 32
9y+11-2y = 32
7y+11 = 32
7y = 32-11
7y = 21
y = 21/7
y = 3 is the number of FGs scored
Use this to find z
z = 11-2y
z = 11-2(3)
z = 11-6
z = 5 is the number of PATs scored
Now let's find x
x = y+3
x = 3+3
x = 6 is the number of TDs scored
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Checking the answers:
x+y+z = 6+3+5 = 9+5 = 14 times they scored
6x+3y+z = 6*6+3*3+5 = 36+9+5 = 50 points scored
There are 3 more TDs than FGs since x = y+3 leads to 6 = 3+3 which is a true statement.
All three original equations we set up for the system of equations are true. Therefore, the final answers are confirmed.