Question

Define operation $\heartsuit$ so that \[ a \heartsuit b = \dfrac{a + b}{2}. \] Let $H$ be a function defined by \[H(x,y,z) = \left(x \heartsuit y\right) \heartsuit z - x \heartsuit \left(y \heartsuit z\right). \] Write an expression for a simplified version of this function: $H(x,y,z) = \boxed{\phantom{blaaaaaaaaaa^2_2}}$ Write an expression, not an equation. Sample answer: $x + y + z$

Answer 1

(-x+z)/4

Explanation:

Evaluate the $\heartsuit$ operation to simplify.

\begin{align*}

H(x,y,z) &= \left(x \heartsuit y\right) \heartsuit z - x \heartsuit \left(y \heartsuit z\right) \\

&= \left(\dfrac{x+y}{2}\right) \heartsuit z - x \heartsuit \left(\dfrac{y+z} 2\right) \\

&= \dfrac{\dfrac{x+y}{2} + z}{2} - \dfrac{x + \dfrac{y+z}{2}}{2}

\end{align*}This looks ugly, but it works out! Continue by multiplying each fraction's numerator and denominator by $2$, in order to eliminate the complex fractions.

\begin{align*}

H(x,y,z) &= \dfrac{\dfrac{x+y}{2} + z}{2} - \dfrac{x + \dfrac{y+z}{2}}{2} \\

&= \dfrac{x + y + 2z}{4} - \dfrac{2x + y + z}{4} \\

&= \dfrac{(x+y+2z) - (2x + y + z)}{4}

&= \dfrac{-x+z}{4}

\end{align*}Surprisingly it simplifies all the way down to $H(x,y,z) = \boxed{\dfrac{-x + z}{4}}$. The value of $y$ is not even used!