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Chapter 7: Problem 80

$$ \frac{3 s^{2}}{3 s-2}+\frac{13 s-10}{3 s-2} $$

Short Answer

Expert verified
\[ \frac{3s^2 + 13s - 10}{3s-2} \]

Step by step solution

01

Combine the Fractions

Since the denominators are the same, combine the numerators over the common denominator: \[ \frac{3s^2}{3s-2} + \frac{13s-10}{3s-2} = \frac{3s^2 + 13s - 10}{3s-2} \]
02

Simplify the Numerator

Combine like terms in the numerator: \[ 3s^2 + 13s - 10 \]
03

Write the Final Expression

Place the simplified numerator over the common denominator to get the final simplified expression: \[ \frac{3s^2 + 13s - 10}{3s-2} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Combining Fractions
Combining fractions is a key skill when working with rational expressions. When the denominators are the same, you can add or subtract the numerators directly. In our example, both fractions have the same denominator, \(3s - 2\). This allows us to combine the fractions by simply adding their numerators. The process can be summarized as follows:
  • Keep the common denominator.
  • Add the numerators together.
This step simplifies the work and makes it easier to handle more complex expressions.
Common Denominator
Finding a common denominator is crucial in the process because it allows you to combine fractions. A common denominator is a shared multiple of the denominators of each fraction involved in the operation. In our exercise, the fractions already have the common denominator \(3s - 2\). When fractions do not have the same denominator, follow these steps to find one:
  • Identify the least common multiple (LCM) of the denominators.
  • Rewrite each fraction so that they both have this LCM as their new denominator.
  • Adjust the numerators accordingly to reflect the changes made to the denominators.
Having a common denominator allows you to easily combine and simplify fractions, as seen in our example.
Simplifying Algebraic Expressions
Simplifying algebraic expressions involves combining like terms and reducing fractions to their simplest form. In the numerator of our combined fraction, we have:
\[3s^2 + 13s - 10\]
We check for any like terms and factor if possible. While simplifying, keep an eye out for opportunities to factorize the expression, which might reveal any further simplifications. Here are the steps:
  • Combine like terms, if any.
  • Look for any common factors in the terms.
  • Factorize the expression, if possible.
No more simplification is needed for our example, so the final expression remains: \[ \frac{3s^2 + 13s - 10}{3s-2} \]
Ensuring the expression is fully simplified makes your answer clean and easier to understand.

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Most popular questions from this chapter

A new energy drink advertises 106 calories for 8 ounces. How many calories are in 12 ounces of the drink?

Solve each rational equation. $$\frac{t}{6 t-12}-\frac{5}{2 t+10}=\frac{t^{2}-23 t+70}{12 t^{2}+36 t-120}$$

Paula and Yuki are roommates. It takes Paula 3 hours to clean their apartment. It takes Yuki 4 hours to clean the apartment. The equation \(\frac{1}{3}+\frac{1}{4}=\frac{1}{t}\) can be used to find \(t,\) the number of hours it would take both of them, working together, to clean their apartment. Explain how this equation models the situation.

If \(a\) varies inversely with \(b\) and \(a=12\) when \(b=\frac{1}{3},\) find the equation that relates \(a\) and \(b\).

Solve. $$\frac{6}{x}+\frac{2}{3}=\frac{1}{y} \text { for } y$$
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