Simplify. Express your answer as a single fraction in simplest form.
$9 - \frac{p}{p-9}$
The expression provided is \(-\frac{p}{p-9}\). Our goal is to simplify this expression and represent it as a single fraction in its simplest form. A single fraction is considered in its simplest form when there are no common factors between the numerator and the denominator, and the signs are handled consistently to represent the expression clearly.
The expression \(-\frac{p}{p-9}\) has a negative sign outside the fraction. We can distribute this negative sign into the denominator to potentially simplify the appearance. Recall that \(- \frac{a}{b} = \frac{a}{-b}\). Applying this property to our expression:
\[-\frac{p}{p-9} = \frac{p}{-(p-9)}\]
Distributing the negative sign inside the parentheses in the denominator gives:
\[-(p-9) = -p + 9 = 9-p\]
Substituting this back into the fraction, we get:
\[\frac{p}{9-p}\]
Both forms \(-\frac{p}{p-9}\) and \(\frac{p}{9-p}\) are single fractions. However, \(\frac{p}{9-p}\) is often preferred as it avoids the negative sign in front of the fraction, making the expression more compact. Since there are no common algebraic factors between \(p\) and \(9-p\), this is the simplest form of the given expression. Thus, the final simplified form is \(\frac{p}{9-p}\).
Simplifying expressions like these is essential in fields such as engineering or physics where you often manipulate formulas to analyze the behavior of systems. For example, if you are modeling the velocity of an object and arrive at an expression like \(-\frac{p}{p-9}\) where \(p\) represents time, rewriting it as \(\frac{p}{9-p}\) can help you immediately identify the vertical asymptote at \(p=9\) and understand the system's behavior as it approaches this point. These algebraic skills provide the foundation for simplifying more complex models and solving differential equations.