Laura Farnam

Linear Thinking:Solving First Degree

Equations

First Degree EquationsTechniques developed

wherever math was studied

Rhind PapyrusA quantity; its half and

its third are added to it. It becomes 10.

x + (1/2)x + (1/3)x = 10

False PositionDouble False Position

False PositionGuessing methodExpect wrong

answerMake computations

easy

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 = 154 + (1/4)(4) = 4+1 = 515/5 = 34 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 = 4615 + 5 + 3 = 2346/23 = 215 x 2 = 30

False Position

False Position

Why Does it Work?Ax = BMultiply x by a factorA(kx) = k(Ax) = kB

Double False PositionUsed in textbooks

until 19th centuryDaboll’s

Schoolmaster’s Assistant (early 1800s)

Guessing method

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?

Double False Position

StepsGuessFind errorRepeatCross-multiply guesses and errorsTake the difference (if similar) or the sum (if

different)Divide by the difference/sum of the errors

Double False Position

Double False Position

y = mx +bPlot the line using

two points“Rise over Run”

100 – 70 = 100 – 80 x – 6 x - 8

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”

Conclusion