Simplify. Express your answer as a single fraction in simplest form.
To simplify the expression \(-\frac{z}{10z-1}\), we first analyze the numerator and the denominator. The numerator is \(-z\) and the denominator is \(10z-1\). Since there are no common factors between the numerator and the denominator, the expression is already in its simplest form. However, we can express it in a slightly cleaner way by distributing the negative sign.
We have the fraction \(-\frac{z}{10z-1}\). To remove the negative sign from the front of the fraction, we can multiply both the numerator and the denominator by \(-1\):
\[ -\frac{z}{10z-1} = \frac{-1 \cdot z}{-1 \cdot (10z - 1)} = \frac{-z}{-10z + 1} = \frac{-z}{1-10z} \]
Alternatively, if we keep the negative in the numerator, we have \(\frac{-z}{10z-1}\). Both forms are technically simple, but \(\frac{z}{1-10z}\) is a common way to write it to avoid the leading negative. Let's stick with the form that avoids the leading negative sign for simplicity.
\[ \frac{z}{1-10z} \]
Checking the original request, it asks to express as a single fraction. The result \(\frac{z}{1-10z}\) fulfills this requirement.
Simplifying rational expressions is a fundamental skill in algebra, useful for solving equations, graphing functions, and performing calculations in physics or engineering. For example, when analyzing the electrical resistance of circuits or the speed-time relationship of an object, you often encounter complex ratios that need to be simplified to understand the behavior of the system. Reducing these expressions allows for easier manipulation and clearer interpretation of the results in real-world scenarios.