However You Solve It ...
A Wonderful Equation

Let's solve the following equation

(1)
5(2x - 6)

7
=
3(4x - 12)

2

Probably the simplest way to approach solving this equation is to multiply both sides by 14 = 7·2. This leads to

10(2x - 6) = 21(4x - 12).

Multiply through:

20x - 60 = 84x - 252.

Add 20x and then 252 to both sides:

64x = 192.

Finallly, divide by 62 to obtain x = 192/64 = 3.

Ed Barbeau mentions a professor one of whose students multiplied both sides by 2 and forgot the 7, a second multiplied both sides by 7 and forgot the 2, and a third neglected both the 7 and the 2. All got correct answers.

For example, let's assume we forgot to multiply by 2:

 5(2x - 6)= 21(4x - 12).
 10x - 30= 84x - 252.
 74x= 222.
 x= 222/74 = 3.

Just as before. How come?

References

  1. E. Barbeau, FALLACIES, FLAWS, AND FLIMFLAM, College Mathematics Journal, v 38, n 3, 2007, p. 220

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