Problem 3 - Algebra 1

Question

You can use \(x-3\) as the LCD when finding the sum \(\frac{5}{x-3}+\frac{2}{3-x}\) What number can you multiply the numerator and the denominator of the second fraction by to get an equivalent fraction with \(x-3\) as the new denominator?

Step-by-Step Solution

Answer

-1

Step 1: Transformation of the denominator

A first step in solving this problem is transforming the \(-3+x\) term (that makes it the additive inverse of \(x-3\)) to become \(x-3\) without changing the mathematical essence. You should focus on how the sign changes to positive when we multiply with a negative number.

Step 2: Find the number that converts the denominator to \(x - 3\)

It can be noticed that multiplying \(-3+x\) or \(3-x\) by \(-1\), the result will be \(x-3\). Thus, \(-1\) is the number that makes the denominator \(x-3\).

Step 3: Apply the same operation to the numerator

To keep the fraction equivalent, the same operation (multiplication by -1) must be applied to the numerator of the fraction. Therefore, the new numerator will be \(-2\).

Key Concepts

Fractions

Fractions are a way to represent parts of a whole. They consist of two numbers: a numerator on top, and a denominator on the bottom. The numerator indicates how many parts you have, while the denominator shows how many parts make up the whole. Understanding fractions is crucial in algebra, especially when dealing with algebraic expressions and equations.

In our exercise, we have two fractions: \( \frac{5}{x-3} \) and \( \frac{2}{3-x} \). To add these fractions, they need a common denominator. This common denominator, often referred to as the Least Common Denominator (LCD), allows you to combine fractions into a single fraction easily. In this exercise, the challenge is to express both denominators as the same term: \( x-3 \). By transforming the fractions to have the same denominator, we make them compatible for addition.

Some key aspects of working with fractions include:

• Finding a common denominator when adding or subtracting fractions
• Maintaining equivalent values by performing the same mathematical operations on both the numerator and the denominator
• Ensuring the mathematical equivalence when manipulating fractions

Algebraic expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operations like addition, subtraction, multiplication, and division. Variables such as \(x\) are used to represent unknowns and can be manipulated in expressions and equations.

In this particular exercise, both fractions have variables in their denominators: \(x-3\) and \(3-x\). To solve the problem, we must manipulate the algebraic expression of the second fraction's denominator to match the first one's denominator. This involves recognizing that expressions like \(-3+x\) and \(3-x\) can be rewritten through simple manipulations such as multiplying by \(-1\), which changes the signs and gives us the form \(x-3\).

Here are a few points to remember about algebraic expressions:

• Variables can represent numbers and are critical for solving equations
• Understanding how to manipulate expressions is key in algebra
• Sign changes happen by operations like multiplication with negative one

Equivalent fractions

Equivalent fractions are different representations of the same fraction. They are obtained by multiplying or dividing both the numerator and the denominator by the same non-zero number. This principle is crucial for maintaining the value of a fraction while changing its appearance.

In the exercise, to make \(\frac{2}{3-x}\) equivalent to a fraction with a denominator of \(x-3\), multiply both the numerator and the denominator by \(-1\). This yields the equivalent fraction \(\frac{-2}{x-3}\). By doing this, the fractions \(\frac{5}{x-3}\) and \(\frac{-2}{x-3}\) now have the same denominator and can therefore be added together.

Keep the following points in mind when working with equivalent fractions:

• Multiplying or dividing both parts of the fraction by the same number keeps their value unchanged
• Equivalent fractions are key for operations like adding or subtracting fractions
• The concept is essential to make sure fractions can be combined effectively