Can You Find The Value Of The Expression?

Algebra Level 3

A A and B B are real numbers such that A = 2 B − 1 , B ≠ 1. A = \frac{2}{B-1}, \quad B \neq 1. What is 2 A − B ( 1 − A ) − ( A + 3 ) ? \frac{2}{A} - B(1-A)-(A+3)?

-1 4 2 0 -2

Calvin Osborne by Calvin Osborne , 18, USA

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1 solution

Calvin Osborne
May 15, 2018

2 A − B ( 1 − A ) − ( A + 3 ) = 2 A − B + A B − A − 3 [because A B − A = 2 ] = 2 A − B + 2 − 3 = 2 − A B − A A [because 2 = A B − A ] = A B − A − A B − A A = − 2 A A = − 2 \begin{aligned} \frac{2}{A} - B(1-A)-(A+3) &= \frac{2}{A} - B +AB - A - 3 \qquad \small \color{#3D99F6} \text{[because } AB - A = 2] \\ &= \frac{2}{A} - B + 2 - 3 \\ &= \frac{2 - AB - A}{A} \qquad \qquad \quad \space \space \space \small \color{#3D99F6} \text{[because } 2 = AB - A] \\ &= \frac{AB - A - AB - A}{A} \\ &= \frac{-2A}{A} = \boxed{-2} \end{aligned}

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