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Chapter 7: Problem 11

Add or subtract. Write answer in lowest terms. \(\frac{5}{y+4}-\frac{1}{y+4}\)

Short Answer

Expert verified
\frac{4}{y+4}

Step by step solution

01

Identify the Common Denominator

Notice that both fractions \(\frac{5}{y+4}\) and \(\frac{1}{y+4}\) have the same denominator, \(y+4\).
02

Combine the Numerators

Since the denominators are the same, combine the numerators by subtraction: \(\frac{5-1}{y+4}\).
03

Simplify the Expression

Perform the subtraction in the numerator: \(\frac{4}{y+4}\).

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Key Concepts

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

Common Denominator
When dealing with fractions, finding a common denominator is crucial. A common denominator is a shared multiple of the denominators of two or more fractions. For example, in the fractions \(\frac{5}{y+4}\) and \(\frac{1}{y+4}\), the denominator is already the same: \(y+4\). This common denominator allows us to combine the fractions easily. If the denominators were different, you would need to find a common multiple. Common denominators make it possible to add or subtract fractions without changing their values.
Fraction Subtraction
Subtracting fractions involves a few key steps. First, ensure that the denominators are the same. If the denominators are not the same, find the least common denominator and convert each fraction to an equivalent one with this new common denominator. In our example, \(\frac{5}{y+4}-\frac{1}{y+4}\), both fractions have the common denominator \(y+4\).
Next, perform the operation on the numerators while keeping the denominator unchanged. Here, the operation is subtraction:
\(\frac{5-1}{y+4}\)
We subtract the numerators (5 - 1) to get 4. The denominator remains \(y+4\), resulting in the fraction \(\frac{4}{y+4}\).
Simplification
Simplification involves making a fraction as simple as possible. For \(\frac{4}{y+4}\), there are no common factors between 4 and \(y+4\), so this fraction is already in its simplest form. However, in general, you should do the following:
  • Divide the numerator and the denominator by their greatest common divisor (GCD).
  • Check if the fraction reduces to an integer or another simple form.
If the numerator and the denominator share a common factor, divide both by this number. Simplifying fractions makes them easier to work with and understand.

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Most popular questions from this chapter

Solve each equation for \(k\) $$ 25=9 k $$

Multiply or divide. Write each answer in lowest terms. See Examples \(3,6,\) and 7 . $$\frac{3 t^{2}-4 t-4}{3 t^{2}+10 t+8} \div \frac{9 t^{2}+21 t+10}{3 t^{2}+14 t+15} $$

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

Simplify each fraction. $$ \frac{1+t^{-1}-56 t^{-2}}{1-t^{-1}-72 t^{-2}} $$

Simplify by starting at "the bottom" and working upward. $$ 5+\frac{5}{5+\frac{5}{5+5}} $$
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