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CEMC Cayley, 2022

Algebra Difficulty 2.3 Junior Find the answer Canada

The operation a∇ba \nabla b is
defined by a∇b=a+ba−ba\nabla b = \dfrac{a+b}{a-b} for all integers aa and bb with a≠ba \neq b. For example, 2∇3=2+32−3=−52\nabla 3 = \dfrac{2+3}{2-3} = -5. If 3∇b=−43\nabla b=-4, what is the value of bb?

Pick one

Solution

Using the definition, 3∇b=3+b3−b3\nabla b = \dfrac{3+b}{3-b}.

Assuming b≠3b \neq 3, the following
equations are equivalent: 3∇bamp;=−43+b3−bamp;=−43+bamp;=−4(3−b)3+bamp;=−12+4b15amp;=3b\begin{align*} 3 \nabla b & = -4 \\ \dfrac{3+b}{3-b} & = -4 \\ 3+b & = -4(3-b) \\ 3+b & = -12 + 4b \\ 15 & = 3b\end{align*} and so b=5b = 5.

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Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.