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Chapter 8: Problem 10

Add or subtract, as indicated. $$\frac{5}{z-3}+\frac{2 z}{z-3}$$

Short Answer

Expert verified
The short answer to the question is: \(\frac{5}{z-3}+\frac{2 z}{z-3} = \frac{5 + 2z}{z-3}\)

Step by step solution

01

Identify the Common Denominator

Both fractions have the same denominator, which is \((z - 3)\). We won't need to find a Least Common Denominator (LCD) in this case because the denominators are already the same.
02

Add the Numerators

Add the numerators of the two fractions together and keep the common denominator. The expression becomes: \(\frac{5}{z-3}+\frac{2z}{z-3} = \frac{5 + 2z}{z-3}\)
03

Simplify the Result, If Possible

There is no common factor between the numerator \(5+2z\) and the denominator \((z-3)\). Therefore, the result cannot be simplified further. So, the final answer is: \(\frac{5}{z-3}+\frac{2 z}{z-3} = \frac{5 + 2z}{z-3}\)

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Key Concepts

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

Common Denominator
Fractions can only be added or subtracted when they have the same denominator, which is known as having a 'common denominator'. The denominator is the bottom part of the fraction, and it tells us how many equal parts the whole is divided into. In some cases, fractions have different denominators, and you must find a common denominator by determining the Least Common Denominator (LCD) to proceed with addition or subtraction.
In this particular exercise, both fractions already have the same denominator, which is \( (z - 3) \). This makes the operation easier as no additional steps are needed to find and adjust to a common denominator.
  • Identify whether the denominators are the same.
  • If they aren't, convert them to a common denominator using the LCD.
  • Ensure all denominators match before performing addition or subtraction.
Numerator Addition
Once you confirm that fractions have the same denominator, it's time to add their numerators. The numerator is the top part of a fraction, representing how much of the whole we are considering. Adding numerators with a common denominator remains straightforward and involves only basic arithmetic.
In this exercise, you simply add the numerators \(5\) and \(2z\) to get \(5 + 2z\). The denominator \((z - 3)\) stays the same, as it's common for both fractions. This process is crucial for maintaining the integrity of the fraction by keeping the parts of the whole consistent.
  • Add only the numerators while the same denominator persists.
  • Ensure the operation aligns mathematically by double-checking your numbers.
  • The result from the numerator addition will form the new numerator in your answer.
Algebraic Expression Simplification
The final step in combining fractions is simplifying the resulting algebraic expression if possible. Simplification involves reducing numbers or terms to their simplest form, such as factoring terms or cancelling out common terms. However, not all expressions simplify further.
In this instance, the expression \(\frac{5 + 2z}{z-3}\) does not simplify because there are no common factors between the numerator \(5 + 2z\) and the denominator \((z-3)\). Understanding when an expression can't be simplified helps avoid needless computations.
  • Check for common factors in the numerator and denominator.
  • Reduce expressions where possible by cancelling shared terms.
  • Recognize when an expression is already in its simplest form.

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Most popular questions from this chapter

Solve each equation. $$\frac{b}{b^{2}+b-6}+\frac{3}{b^{2}+9 b+18}=\frac{8}{b^{2}+4 b-12}$$

Determine the domain of each rational function. $$k(t)=\frac{t}{t^{2}-14 t+33}$$

Determine the domain of each rational function. $$g(t)=\frac{3 t-4}{t^{2}-9 t+8}$$

Solve each equation. $$\frac{u}{8}=\frac{2}{10-u}$$

Perform the indicated operations. $$\frac{n}{n^{2}+11 n+30}-\frac{3}{n^{2}+10 n+25}$$
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