# 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

False Position Guessing method Expect wrong answer Make computations easy False Position

Example 1: A quantity; its fourth is added to it. It becomes 15. 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 False Position Example 2: A quantity; its third and its fifth are added to it. It becomes 46.

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

False Position Why Does it Work? Ax = B Multiply x by a factor A(kx) = k(Ax) = kB Double False Position Used in textbooks

until 19th century Dabolls Schoolmasters Assistant (early 1800s) Guessing method 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 ones share of the money? 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

Double False Position y = mx +b Plot the line using two points Rise over Run 100 70 = 100 80

x6 x-8 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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