/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 61 Simplify each expression. $$ ... [FREE SOLUTION] | 91影视

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Chapter 14: Problem 61

Simplify each expression. $$ \frac{3}{a-2}+\frac{2}{a-3} $$

Short Answer

Expert verified
The simplified expression is \( \frac{5a - 13}{(a-2)(a-3)} \).

Step by step solution

01

Identify the Problem

We need to simplify the expression \( \frac{3}{a-2} + \frac{2}{a-3} \). This expression is a sum of two fractions with different denominators.
02

Find a Common Denominator

The denominators \( a-2 \) and \( a-3 \) are different, so find the least common denominator (LCD). The LCD of \( a-2 \) and \( a-3 \) is their product \((a-2)(a-3)\).
03

Rewrite Each Fraction

Rewrite each fraction with the common denominator. This means:\[\frac{3}{a-2} = \frac{3(a-3)}{(a-2)(a-3)} \] and \[\frac{2}{a-3} = \frac{2(a-2)}{(a-2)(a-3)} \]
04

Add the Fractions

Add the fractions by combining their numerators since they now have the same denominator:\[\frac{3(a-3)}{(a-2)(a-3)} + \frac{2(a-2)}{(a-2)(a-3)} = \frac{3(a-3) + 2(a-2)}{(a-2)(a-3)}\]
05

Simplify the Numerator

Distribute and combine like terms in the numerator:\[3(a-3) = 3a - 9\] and \[2(a-2) = 2a - 4\]Thus, \[3a - 9 + 2a - 4 = 5a - 13\]
06

Final Expression

The simplified expression is:\[\frac{5a - 13}{(a-2)(a-3)}\]

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Key Concepts

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

Least Common Denominator
When dealing with the addition of fractions, it's essential to have a common denominator. The Least Common Denominator (LCD) helps us to reframe the fractions with a shared base, allowing them to be combined. Let's delve into some strategies to find the LCD:
  • The LCD is the smallest expression that both denominators can divide into without leaving a remainder.
  • The simplest approach is to multiply the two denominators together. In this exercise, the denominators are \(a-2\) and \(a-3\), so their product \((a-2)(a-3)\) becomes the LCD.
  • Use the LCD to transform each fraction, allowing them to have the same denominator. This step is crucial for subsequent addition.
Finding the LCD effectively sets the stage for combining the fractions. It simplifies the process and makes the addition straightforward.
Fraction Addition
Adding fractions is straightforward once they share a denominator. It involves combining the numerators while retaining the common denominator. Here's how it works:
  • After rewriting each fraction with the LCD identified earlier, focus on the numerators.
  • In this exercise, transform the fractions \(\frac{3}{a-2}\) and \(\frac{2}{a-3}\) to have a common denominator \((a-2)(a-3)\).
  • The fractions then become \(\frac{3(a-3)}{(a-2)(a-3)}\) and \(\frac{2(a-2)}{(a-2)(a-3)}\).
With the same denominators, add the numerators keeping the denominator unchanged:\[\frac{3(a-3)}{(a-2)(a-3)} + \frac{2(a-2)}{(a-2)(a-3)} = \frac{3(a-3) + 2(a-2)}{(a-2)(a-3)}\]This step ensures your fraction is properly combined before simplifying.
Expression Simplification
Simplifying an expression means reducing it to its simplest form. Here鈥檚 a step-by-step of how to simplify the resulting expression from adding fractions:
  • First, distribute any factored terms in the numerator. For example, distribute \(3\) across \(a-3\) and \(2\) across \(a-2\).
  • This results in: \(3(a-3) = 3a - 9\) and \(2(a-2) = 2a - 4\).
  • Combine like terms by adding the results: \(3a - 9 + 2a - 4 = 5a - 13\).
Your simplified numerator now becomes \(5a - 13\). Keep the denominator \((a-2)(a-3)\) as it is, as it cannot be simplified further. Thus, the final simplified expression is:\[\frac{5a - 13}{(a-2)(a-3)}\]This final form is concise and ready for interpretation or further calculations. Simplification is key in making expressions manageable and comprehensible.

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Most popular questions from this chapter

Solve each equation for all values of \(\theta\). \(2 \sin ^{2} \theta-3 \sin \theta-2=0\)

Find all solutions of each equation for the given interval. \(2 \cos \theta-1=0 ; 0^{\circ} \leq \theta<360^{\circ}\)

Find the exact value of each expression. \(\sin \left(-135^{\circ}\right)\)

Solve each equation for all values of \(\theta\) if \(\theta\) is measured in degrees. \(\sin ^{2} \theta-2 \sin \theta-3=0\)

PREREQUISITE SKILL Solve each equation. $$ (x+6)(x-5)=0 $$
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