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Chapter 7: Problem 24

For exercises 1-66, simplify. $$ \frac{5 x-15}{9 x-27} $$

Short Answer

Expert verified
\(\frac{5}{9}\)

Step by step solution

01

Factor the numerator

Factor out the greatest common factor (GCF) of the terms in the numerator. The numerator is \(5x - 15\), and the GCF is 5. So, \(5x - 15\) can be written as \(5(x - 3)\).
02

Factor the denominator

Factor out the GCF of the terms in the denominator. The denominator is \(9x - 27\), and the GCF is 9. So, \(9x - 27\) can be written as \(9(x - 3)\).
03

Simplify the fraction

Now the fraction is \(\frac{5(x - 3)}{9(x - 3)}\). Since \(x - 3\) is a common factor in both the numerator and the denominator, they cancel each other out. The simplified fraction is \(\frac{5}{9}\).

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Key Concepts

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

greatest common factor
In algebra, finding the greatest common factor (GCF) of terms is crucial for simplifying expressions. The GCF is the largest number that divides all the terms completely. For example, in the numerator of our exercise, which is \(5x - 15\), the GCF is 5. This is because both 5 and 15 can be divided by 5.
To find the GCF, list the factors of each term and identify the highest common factor:
  • Factors of 5x: 1, 5, x, 5x
  • Factors of 15: 1, 3, 5, 15
(The factor we can see here is 5)
We use this factor to rewrite the expression by factoring it out, giving us \(5(x - 3)\). Similarly, for the denominator \(9x - 27\), the GCF is 9, which means we factor it like this: \(9(x - 3)\).
Extracting the GCF simplifies the process of breaking down and simplifying fractions.
factoring expressions
Factoring expressions involves breaking down complex terms into simpler factors. This makes it easier to simplify and solve equations.
In our example, we need to factor both the numerator and the denominator.
  • For the numerator \(5x - 15\), we factor out the greatest common factor (GCF), which is 5, resulting in \(5(x - 3)\).
  • For the denominator \(9x - 27\), we factor out the GCF, 9, resulting in \(9(x - 3)\).
Now, the fraction is:\( \frac{5(x - 3)}{9(x - 3)} \). Recognize the repeated factor, \(x - 3\), in both terms.
Factoring brings terms into a form that makes further simplification and cancellations possible.
simplification
Simplification is about reducing expressions to their simplest form. It's the last step where we want to make everything as clean and concise as possible.

In our exercise, after factoring both the numerator and the denominator, we get: \(\frac{5(x - 3)}{9(x - 3)}\). Notice that \(x - 3\) appears in both the numerator and the denominator. These terms cancel each other out, leaving us with the simplified fraction \( \frac{5}{9} \).
The cancellation works because dividing any number by itself equals 1 (\( \frac{(x - 3)}{(x - 3)} = 1 \)). Keep in mind:
  • Always factor expressions entirely.
  • Identify and cancel common factors in fractions.
Simplifying makes expressions easier to understand and work with.

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Most popular questions from this chapter

The relationship of \(x\) and \(y\) is a direct variation. When \(x=1, y=6\). a. Find the constant of proportionality, \(k\). b. Write an equation that represents this direct variation. c. Find \(y\) when \(x=4\), d. Use slope-intercept graphing to graph this equation. e. Use the graph to find \(y\) when \(x=2\).

For exercises \(35-36, T=\frac{336 \mathrm{gm}}{R}\) represents the relationship of tire diameter, \(T\); gear ratio, \(g\); speed, \(m\); and revolutions of the tire per minute, \(R\). Is the relationship of the given variables a direct variation or an inverse variation? $$ g \text { and } R \text { are constant; the relationship of } T \text { and } m \text {. } $$

For exercises 31-40, (a) solve. (b) check. $$ \frac{3}{w-3}+\frac{4}{w}=\frac{w}{w-3} $$

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For a fixed number of hotel rooms, the number of rooms cleaned per hour, \(x\), and the number of hours it takes to clean the rooms, \(y\), is an inverse variation. If a person can clean 8 rooms per hour, it takes 15 hr to clean the rooms. a. Find the constant of variation, \(k\). Include the units of measurement. b. Write an equation that represents this relationship. c. If a person can clean 6 rooms per hour, find the time needed to clean the rooms.
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