Problem 2 - Elementary Algebra

Question

For the following problems, fill in the missing term. -\frac{3}{x+4}=\frac{\underline{\phantom{xx}}}{x+4}$

Step-by-Step Solution

Answer

Question: Given the fraction -3/(x+4) and another fraction with a missing term in the numerator and (x+4) in the denominator, find the missing term so that both fractions are equal. Answer: -3

Step 1: Understand the problem

We are given two fractions. The first fraction has -3 in the numerator and (x+4) in the denominator. The second fraction has a missing term in the numerator and (x+4) in the denominator. Our task is to find the missing term so that both fractions are equal.

Step 2: Identify the relationship between both fractions

Since both fractions have the same denominator (x+4), it means they should be equal if their numerators are equal too.

Step 3: Write down the equation using the condition for the fractions to be equal

Since the condition for both fractions to be equal is that their numerators must be equal, let's represent the missing term with the symbol "a". Then, we have the equation as follows: -\frac{3}{x+4}=\frac{a}{x+4}

Step 4: Solve for the missing term

Since both fractions have the same denominator (x+4), we can write the equation as -3 = a. From this, we can see that a = -3.

Step 5: Write the complete fraction with the missing term filled in

Now that we have found the missing term, we can write the fraction as -\frac{3}{x+4} = \frac{-3}{x+4}. We have successfully filled in the missing term.

Key Concepts

Fractions

Fractions are a way to represent part of a whole or a division of quantities. They consist of two main parts: the numerator and the denominator, separated by a line. A fraction helps in dividing something into equal parts.

When dealing with fractions, the numerator is the top number, which represents how many parts you have. For instance, in the fraction \( \frac{3}{4} \), the numerator is 3.

The denominator is the bottom number of the fraction, indicating the total number of equal parts the whole is divided into. In \( \frac{3}{4} \), the denominator is 4, which means the whole is divided into 4 equal parts.

• A proper fraction has a numerator smaller than its denominator.
• An improper fraction has a numerator larger than its denominator.
• If the numerator is zero, the entire fraction equals zero.
Understanding these basics helps in both forming and manipulating fractions effectively.

Numerators

The numerator is the top component of a fraction and plays a crucial role in defining the value of the fraction. It can be any integer, positive or negative, depending on the context of the problem.

The numerator determines how many parts of the whole (denoted by the denominator) are being considered. In a fraction like \( \frac{-3}{8} \), the numerator -3 indicates that there are 3 parts taken away from the whole represented by 8 parts.

• Changing the numerator directly affects the fraction's value, either increasing or decreasing it.
• If you multiply the numerator by a number, the entire fraction scales by that number but the ratio remains the same if both numerator and denominator are multiplied by the same factor.
• In the equation \(-\frac{3}{x+4} = \frac{a}{x+4} \), the numerators -3 and \(a\) must be equal for the fractions to be equal, as they share the same denominator.

Making sense of numerators and their impact is key for mastering fraction problems and creating equivalent expressions.

Equivalent Fractions

Equivalent fractions are different fractions that represent the same quantity or value. They are a powerful concept in algebra that allow us to simplify, compare, and manipulate fractions across different problems.

Two fractions are considered equivalent if their cross-products are equal, or simply when their reduced forms are identical.

• For example, \( \frac{1}{2} \) and \( \frac{2}{4} \) are equivalent since they both simplify to \( \frac{1}{2} \).
• Fractions with the same numerator and denominator are always equal.
• To find equivalent fractions, multiply or divide both the numerator and the denominator by the same nonzero number.

In our exercise, finding an equivalent fraction meant finding the numerator that makes both fractions \(-\frac{3}{x+4}\) and \(\frac{a}{x+4}\) equal, leading to \(a = -3\).
Understanding equivalent fractions unlocks the ability to solve equations and make different fractions comparable.