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Chapter 7: Problem 58

Add or subtract. Write answer in lowest terms. \(\frac{t}{8 t-9 s}-\frac{s}{9 s-8 t}\)

Short Answer

Expert verified
\( \frac{t + s}{8t - 9s} \)

Step by step solution

01

Identify the fractions

First, identify the given fractions: \( \frac{t}{8t - 9s} \) and \( \frac{s}{9s - 8t} \).
02

Express the denominators similarly

Notice that \( 9s - 8t = -(8t - 9s) \), hence we can rewrite the second fraction as: \( \frac{s}{9s - 8t} = -\frac{s}{8t - 9s} \).
03

Simplify the expression

Using the above step, rewrite the expression: \( \frac{t}{8t - 9s} - \frac{s}{9s - 8t} = \frac{t}{8t - 9s} + \frac{s}{8t - 9s} \).
04

Combine the fractions

Now that the denominators are the same, combine the fractions: \( \frac{t}{8t - 9s} + \frac{s}{8t - 9s} = \frac{t + s}{8t - 9s} \).
05

Write the final answer in lowest terms

Since \( \frac{t + s}{8t - 9s} \) is already in its simplest form, the final answer is \( \frac{t + s}{8t - 9s} \).

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Key Concepts

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

Addition of Fractions
When adding fractions, the key is to have a common denominator. The denominator is the bottom part of a fraction, and it shows into how many parts the whole is divided.
For instance, adding \(\frac{3}{4}\) and \(\frac{2}{4}\) is straightforward because both fractions have the same denominator鈥�4. You simply add the numerators (the top parts): 3 + 2, giving you \(\frac{5}{4}\).
In more complex cases, like in our exercise, you may need to manipulate the fractions to achieve a common denominator:
  • Identify and rewrite the denominators so they are the same.
  • Once the denominators match, combine the numerators.
Mastering this works as a useful tool not only for simple numbers but also for algebraic expressions.
Subtraction of Fractions
Subtraction of fractions follows a very similar process to addition of fractions; the denominators must be the same. When the denominators differ, you need to find a common base:
For example, to subtract \(\frac{5}{6}\) from \(\frac{7}{6}\), do this:
  • Both fractions already have 6 as the denominator.
  • Subtract the numerators directly: 7 - 5 to get \(\frac{2}{6}\).
In the given algebraic fraction example, we used the identity \((9s - 8t) = -(8t - 9s)\) to rewrite and match the denominators:
Once matching, subtraction becomes a simple process of subtracting the numerators.
Lowest Terms
Writing a fraction in its lowest terms means simplifying it so that the numerator and denominator have no common factors other than 1. For example, the fraction \(\frac{4}{8}\) simplifies to \(\frac{1}{2}\) because both the numerator and the denominator can be divided by 4.
This process involves finding the Greatest Common Divisor (GCD) of the numerator and the denominator and dividing both by that number.
In algebraic fractions, you also need to ensure that the expression is simplified fully:
  • Factorize both the numerator and the denominator if necessary.
  • Cancel out any common factors.
In our earlier solution, \(\frac{t + s}{8t - 9s}\) had no common factors to simplify further, so it was already in its lowest terms.

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Most popular questions from this chapter

Multiply or divide. Write each answer in lowest terms. See Examples \(3,6,\) and 7 . $$\frac{2 m^{2}+7 m+3}{m^{2}-9} \cdot \frac{m^{2}-3 m}{2 m^{2}+11 m+5}$$

Add or subtract as indicated. Write each answer in lowest terms. $$ \frac{11}{6}-\frac{2}{5} $$

Bill Veeck was the owner of several major league baseball teams in the 1950 s and \(1960 \mathrm{s}\). He was known to often sit in the stands and en- joy games with his paying customers. Here is a quote attributed to him: I have discovered in 20 years of moving around a ballpark, that the knowledge of the game is usually in inverse proportion to the price of the seats. Explain in your own words the meaning of his statement. (To prove his point, Veeck once allowed the fans to vote on managerial decisions.)

Multiply or divide. Write each answer in lowest terms. See Examples \(3,6,\) and 7 . $$\frac{r^{2}+r-6}{r^{2}+4 r-12} \div \frac{r+3}{r-1}$$

Rewrite each rational expression with the indicated denominator. $$ \frac{4 r-t}{r^{2}+r t+t^{2}}=\frac{?}{t^{3}-r^{3}} $$
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