For any numbers x and y, x<>y = 2x - y - xy. If x<>y = 0, then which

Official Solution:

For any numbers \(x\) and \(y\), \(x \diamond y = 2x - y - xy\). If \(x \diamond y = 0\), then which of the following CANNOT be \(y\)?

A. 3
B. 2
C. 0
D. \(-\frac{4}{3}\)
E. -2

We must determine which of the answer choices cannot be a value of \(y\).

If \(x \diamond y = 2x - y - xy\) and \(x \diamond y = 0\), we can combine equations to get: \(2x - y - xy = 0\). Because the problem tells us that one value for \(y\) is impossible, one answer choice will make the equation \(2x - y - xy = 0\) false.

Plug in each answer choice and see which makes the equation false.

A. If \(y = 3\), then \(2x - y - xy = 0\) becomes \(2x - 3 - x(3) = 0\). Combine like terms: \(-x - 3 = 0\), so \(x = -3\). Because \(x\) is a variable, it can be equal to any quantity. Therefore, this equation is not false.

B. If \(y = 2\), then \(2x - 2 - x(2) = 0\). The \(x\) terms cancel, leaving \(-2 = 0\). This is false, so B is correct. Double-check by making sure that no other answer yields a false equation.

C. If \(y = 0\), then \(2x - 0 - x(0) = 0\). This simplifies to \(2x = 0\), which is not false.

D. If \(y = -\frac{4}{3}\) then \(2x - (-\frac{4}{3}) - x(-\frac{4}{3}) = 0\). Combine like terms: \(\frac{10}{3}x + \frac{4}{3} = 0\). Isolate the \(x\) term: \(\frac{10}{3}x = -\frac{4}{3}\), so \(x = -\frac{2}{5}\). This is not false.

E. If \(y = -2\), then \(2x - (-2) - x(-2) = 0\). Combine like terms: \(4x +2 = 0\). Isolate the \(x\) term: \(4x = -2\), so \(x = -\frac{1}{2}\). This is not false, so only choice B yields a false equation.

Choice B is the correct answer.

Alternatively, we can solve this problem using algebra. Recall that a value can be impossible for a variable in two ways: if the value of the variable causes a division by 0 or if the value forces the equation to look for the square root of a negative number. Since there are no exponents or roots in this equation, the answer will likely involve dividing by zero. If plugging in a certain value for \(y\) causes a division by zero, we should look for \(y\) in the denominator of a fraction. Solving the equation for \(x\) may give us a fraction in terms of \(y\).

First factor out the \(x\) to get \(x(2 - y) - y = 0\).

Add \(y\) to both sides: \(x(2 - y) = y\).

Divide both sides by \(2 - y\) to get \(x = \frac{y}{2-y}\).

If \(y = 2\), the denominator becomes \(2 - 2 = 0\). Any value that makes a denominator equal to 0 cannot be a valid solution to an equation because dividing by 0 is undefined. Therefore, \(y\) cannot be equal to 2.

Answer: B.