Lecture Notes: Application Problems in Guided Notes

Overview

• Discussed application problems from the guided notes.
• Focused on using algebraic methods to solve problems with unknowns.

Problem 15: Baseball Teams Contract Offer

• Objective: Determine the first team's offer for a pitcher.
• Given:
  • Second team offers double the first team's offer plus an extra million per year.
  • Player seeks $18 million per year.
• Solution:
  • Let X be the first team's offer.
  • Equation: 2X + 1 = 18.
  • Solve:
    • Subtract 1: 2X = 17.
    • Divide by 2: X = 8.5.
  • Conclusion: First team's offer was $8.5 million.

Problem 16A: Heights of Donna and Christina

• Objective: Represent Donna's and Christina's heights.
• Given:
  • Donna is 5 inches taller than Christina.
• Solution:
  • Let X represent Christina's height.
  • Donna's height: X + 5.

Problem 16B: Test Grades of Ariana and John

• Objective: Express John's test grade.
• Given:
  • John's grade is 5 points less than twice Ariana's.
• Solution:
  • Let X be Ariana's grade.
  • John's grade: 2X - 5.

Problem 17: Cutting a Board

• Objective: Find lengths of two pieces.
• Given:
  • Board is 6 feet long.
  • One piece is 3 times longer than the other.
• Solution:
  • Let X be the shorter piece.
  • Longer piece: 3X.
  • Equation: X + 3X = 6.
  • Solve:
    • 4X = 6.
    • X = 1.5 feet.
  • Results: Shorter piece = 1.5 ft, Longer piece = 4.5 ft.

Problem 18: Support Post Heights

• Objective: Determine heights of support posts.
• Given:
  • Deck height is 92 inches.
  • Taller post is 8 inches longer than the shorter one.
• Solution:
  • Let X be the shorter post.
  • Taller post: X + 8.
  • Equation: 2X + 8 = 92.
  • Solve:
    • Subtract 8: 2X = 84.
    • Divide by 2: X = 42 inches.
  • Results: Shorter = 42 in, Taller = 50 in.

Problem 19: Tips in Bills

• Objective: Find the number of $1 and $5 bills.
• Given:
  • Total tips: $137
  • 53 more $1 bills than $5 bills.
• Solution:
  • Method 1: Trial and Error
    • Test different combinations of bills.
    • Approach: Adjust number of $5 bills to match total.
  • Method 2: Algebraic Solution
    • Let X be the number of $5 bills.
    • Number of $1 bills: X + 53.
    • Equation: 5X + (X + 53) = 137.
    • Simplify and solve:
      • 6X + 53 = 137
      • 6X = 84
      • X = 14
    • Results: 14 $5 bills, 67 $1 bills.

Conclusion

• Emphasis on using algebraic equations to simplify problem-solving.
• Highlighted both trial/error and algebraic strategies to approach real-world problems.