91Ó°ÊÓ

Algebraic manipulation is a fundamental step in working with equations, especially multivariable ones. It involves rearranging the equation to express one variable in terms of others. This might include operations like moving terms from one side of the equation to the other, factoring, expanding expressions, or simplifying. These operations help isolate the variable of interest.

In the original exercise, the task was to determine if \( z \) can be expressed as a function of \( x \) and \( y \). This started with algebraically rearranging the equation \( xz^2 + 2xy - y^2 = 4 \) to solve for \( z^2 \). In this case, the rearrangement moves the terms not associated with \( z \) to the other side, producing \( xz^2 = 4 - 2xy + y^2 \), and then dividing by \( x \), we get \( z^2 = \frac{4 - 2xy + y^2}{x} \).

This step is crucial because it simplifies the equation down to a form where \( z \) can be isolated through further manipulation.

Solving equations, especially those involving multiple variables, often requires additional steps beyond just rearrangement. Once you isolate a variable like we did with \( z^2 \) in the previous step, the next move is commonly to solve for that variable completely.

In our exercise, the variable \( z \) was isolated by applying a square root to both sides of the reorganized equation. Solving \( z^2 = \frac{4 - 2xy + y^2}{x} \) involves extracting the square root, yielding \( z = \pm \sqrt{\frac{4 - 2xy + y^2}{x}} \).

This step is critical to solving equations because it provides the potential solutions for the isolated variable. It is also pivotal to consider both positive and negative roots, as square roots can lead to two potential values for \( z \).

• Positive and negative square roots imply multiple potential solutions.
• Checking all possible solutions is key in determining if this can define a function.

Function definition involves understanding the mapping from inputs to outputs. A function, in mathematical terms, is a relation where each input is associated with exactly one output. When checking if \( z \) is a function of \( x \) and \( y \), you want each pair \((x, y)\) to result in a single \( z \) value.

In our problem, after solving the equation, we obtained \( z = \pm \sqrt{\frac{4 - 2xy + y^2}{x}} \). The presence of the \( \pm \) symbol indicates that for each input \((x, y)\), there could be two different \( z \) outputs (one positive and one negative).

This dual output violates the definition of a function since it means \( z \) cannot be a unique output for each \((x, y)\) input. Therefore, it's concluded that \( z \) is not a function of \( x \) and \( y \).

This concept is crucial in multivariable calculus as it defines how inputs relate to potential outputs, distinguishing functional relationships from mere equations.