Let’s solve this step by step.
We are told:
- Total points = 50
- Total scoring plays = 14
- Scoring types:
- Touchdowns (T) = 6 points each
- PATs (P) = 1 point each
- Field goals (F) = 3 points each
- They had
three more touchdowns than field goals → T = F + 3
We need to find how many of each: T, P, and F.
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Step 1: Write equations based on the info.
Equation 1 (total scores):
T + P + F = 14
Equation 2 (total points):
6T + 1P + 3F = 50
Equation 3 (relationship between T and F):
T = F + 3
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Step 2: Substitute Equation 3 into Equations 1 and 2.
Substitute T = F + 3 into Equation 1:
(F + 3) + P + F = 14
→ 2F + P + 3 = 14
→ 2F + P = 11 → Let’s call this Equation A
Now substitute T = F + 3 into Equation 2:
6(F + 3) + P + 3F = 50
→ 6F + 18 + P + 3F = 50
→ 9F + P + 18 = 50
→ 9F + P = 32 → Let’s call this Equation B
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Step 3: Subtract Equation A from Equation B to eliminate P.
Equation B: 9F + P = 32
Equation A: 2F + P = 11
Subtract: (9F + P) - (2F + P) = 32 - 11
→ 7F = 21
→ F = 3
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Step 4: Find T using T = F + 3
T = 3 + 3 = 6
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Step 5: Find P using Equation A: 2F + P = 11
2(3) + P = 11
6 + P = 11
P = 5
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Step 6: Check if everything adds up.
Total scores: T + P + F = 6 + 5 + 3 = 14
✔
Total points: 6*6 + 1*5 + 3*3 = 36 + 5 + 9 = 50
✔
Touchdowns vs field goals: 6 = 3 + 3
✔
Everything checks out!
Final Answer:
Touchdowns: 6, PATs: 5, Field Goals: 3
Parent Tip: Review the logic above to help your child master the concept of systems of equations 3 variables word problems worksheet.
