# 2016 AMC 8 Problems/Problem 10

Suppose that $a * b$ means $3a-b.$ What is the value of $x$ if $$2 * (5 * x)=1$$ $\textbf{(A) }\frac{1}{10} \qquad\textbf{(B) }2\qquad\textbf{(C) }\frac{10}{3} \qquad\textbf{(D) }10\qquad \textbf{(E) }14$

## Solution

Let us plug in $(5 * x)=1$ into $3a-b$. Thus it would be $3(5)-x$. Now we have $2*(15-x)=1$. Plugging $2*(15-x)$ into $3a-b$, we have $6-15+x=1$. Solving for $x$ we have $$-9+x=1$$ $$x=\boxed{\textbf{(D)} \, 10}$$

## Solution 2

Let us set a variable $y$ equal to $5 * x$. Solving for y in the equation $3(2)-y=1$, we see that y is equal to five. By substitution, we see that $5 * x$ = 5. Solving for x in the equation $5(3)-x = 5$ we get $$x=\boxed{\textbf{(D)} \, 10}$$

 2016 AMC 8 (Problems • Answer Key • Resources) Preceded byProblem 9 Followed byProblem 11 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 All AJHSME/AMC 8 Problems and Solutions

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