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Chapter 7: Problem 8

Perform the indicated operation. Simplify, if possible. \(\frac{4}{9 t}-\frac{7}{6 t}\)

Short Answer

Expert verified
\(\frac{4(a - 1)}{7}\)

Step by step solution

01

Identify a common denominator

Both fractions \(\frac{a}{7}\) and \(\frac{3a-4}{7}\) have the same denominator of 7.
02

Add the numerators

Since the denominators are the same, add the numerators while keeping the same denominator: \(\frac{a}{7} + \frac{3a-4}{7} = \frac{a + (3a - 4)}{7}\).
03

Simplify the numerator

Combine like terms in the numerator: \(\frac{a + 3a - 4}{7} = \frac{4a - 4}{7}\).
04

Factor the numerator, if possible

Factor out the common factor in the numerator: \(\frac{4a - 4}{7} = \frac{4(a - 1)}{7}\).
05

Final simplified form

The final simplified form is \(\frac{4(a - 1)}{7}\).

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Key Concepts

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

Common Denominator
When working with fractions, a common denominator is essential. It lets us add or subtract fractions smoothly. In this exercise, we started with two fractions: \(\frac{a}{7}\) and \(\frac{3a-4}{7}\). Since both fractions have the same denominator (7), we could combine them easily. If the denominators were different, we would first need to find the Least Common Denominator (LCD) before proceeding. This step ensures the fractions are compatible for addition or subtraction.
Combining Like Terms
Combining like terms means grouping and simplifying terms that have the same variable parts. After identifying a common denominator, we added the numerators of the fractions together: \(a + (3a - 4)\). Notice that \(a\) and \(3a\) are like terms because they both contain the variable \(a\).

This allowed us to combine them into \(4a - 4\). Be sure to keep constants and variables separate while combining like terms. This simplifies further operations.
Factoring
Factoring involves breaking down an expression into simpler components. After combining like terms, we had \(\frac{4a - 4}{7}\). To simplify this further, we can factor the numerator.

Both terms in the numerator, 4a and -4, have a common factor of 4. So, we factored it out: \(4(a - 1)\).

Factoring can help simplify complex expressions, making it easier to understand and work with.
Fraction Operations
Working with fractions involves several crucial steps. In this exercise, we utilized the common denominator to add fractions.

Here鈥檚 a summary of important steps in fraction operations:
  • Find a common denominator for addition or subtraction. If needed, adjust the fractions to have this denominator.
  • Combine the numerators while keeping the denominator constant.
  • Simplify the numerator by combining like terms.
  • Factor the numerator, if possible, to simplify the fraction further.
  • Always check if your fraction can be reduced for a simpler form.


Mastering these steps will make handling fraction operations much more manageable.

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Most popular questions from this chapter

Solve. $$\frac{5-3 a}{a^{2}+4 a+3}-\frac{2 a+2}{a+3}=\frac{3-a}{a+1}$$

Find an equation of variation in which: \(y\) varies jointly as \(x\) and \(z\) and inversely as \(w,\) and \(y=\frac{3}{2}\) when \(x=2, z=3,\) and \(w=4\)

To prepare for Chapter \(8,\) review solving inequalities \((\text {Section } 2.6)\) Solve. $$ \frac{5-x}{2} \geq 1 $$

Aviation. A Coast Guard plane has enough fuel to fly for \(6 \mathrm{hr},\) and its speed in still air is \(240 \mathrm{mph}\). The plane departs with a 40 -mph tailwind and returns to the same airport flying into the same wind. How far from the airport can the plane travel under these conditions? (Assume that the plane can use all its fuel.)

To prepare for Chapter \(8,\) review solving inequalities \((\text {Section } 2.6)\) Solve. $$ 3-x<3 x+5 $$
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