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Linear Thinking: Solving First Degree Equations

Linear Thinking: Solving First Degree Equations. Laura Farnam. First Degree Equations. Techniques developed wherever math was studied Rhind Papyrus A quantity; its half and its third are added to it. It becomes 10. x + (1/2)x + (1/3)x = 10 False Position Double False Position.

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Linear Thinking: Solving First Degree Equations

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  1. Linear Thinking:Solving First Degree Equations Laura Farnam

  2. First Degree Equations • Techniques developed wherever math was studied • Rhind Papyrus • A quantity; its half and its third are added to it. It becomes 10. • x + (1/2)x + (1/3)x = 10 • False Position • Double False Position

  3. False Position • Guessing method • Expect wrong answer • Make computations easy

  4. False Position • Example 1: A quantity; its fourth is added to it. It becomes 15.

  5. False Position • Example 1: A quantity; its fourth is added to it. It becomes 15. • x + (1/4)x = 15 • 4 + (1/4)(4) = 4+1 = 5 • 15/5 = 3 • 4 x 3 = 12

  6. False Position • Example 2: A quantity; its third and its fifth are added to it. It becomes 46.

  7. False Position • Example 2: A quantity; its third and its fifth are added to it. It becomes 46. • x + (1/3)x + (1/5)x = 46 • 15 + 5 + 3 = 23 • 46/23 = 2 • 15 x 2 = 30

  8. False Position Why Does it Work? • Ax = B • Multiply x by a factor • A(kx) = k(Ax) = kB

  9. Double False Position • Used in textbooks until 19th century • Daboll’s Schoolmaster’s Assistant (early 1800s) • Guessing method

  10. Double False Position • Example 3: A purse of 100 dollars is to be divided among four men A, B, C, and D, so that B may have four dollars more than A, and C eight dollars more than B, and D twice as many as C; what is each one’s share of the money?

  11. Double False Position Steps • Guess • Find error • Repeat • Cross-multiply guesses and errors • Take the difference (if similar) or the sum (if different) • Divide by the difference/sum of the errors

  12. Double False Position • y = mx +b • Plot the line using two points • “Rise over Run” • 100 – 70 = 100 – 80 x – 6 x - 8

  13. Conclusion • False Position (Ax = B) • Double False Position (Ax + C = B) • Linear • A constant ratio • “the change in the output is proportional to the change in the input”

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