Problem 85 - Elementary Algebra

Question

For Problems \(75-92\), perform the indicated operations and express your answers in reduced form. $$ \frac{-5}{4 x^{2}}+\frac{7}{3 x^{2}} $$

Step-by-Step Solution

Answer

\(\frac{13}{12x^2}\).

Step 1: Identifying the Problem

We are given two fractions to add, each having a different numerator but the same denominator of the form \(x^2\). The fractions are \(\frac{-5}{4x^2}\) and \(\frac{7}{3x^2}\). The goal is to combine these fractions and simplify the result.

Step 2: Finding a Common Denominator

Since the fractions have different denominators of \(4x^2\) and \(3x^2\), we need to find their least common denominator (LCD). The LCD is the lowest common multiple of \(4\) and \(3\), which is \(12\). The common denominator will therefore be \(12x^2\).

Step 3: Adjusting the Fractions

To combine the fractions over the common denominator \(12x^2\), we adjust each fraction. We multiply the numerator and denominator of \(\frac{-5}{4x^2}\) by \(3\) to obtain \(\frac{-15}{12x^2}\), and we multiply the numerator and denominator of \(\frac{7}{3x^2}\) by \(4\) to obtain \(\frac{28}{12x^2}\).

Step 4: Adding the Fractions

With the same denominator, we can add the numerators together: \(\frac{-15}{12x^2} + \frac{28}{12x^2} = \frac{-15 + 28}{12x^2} = \frac{13}{12x^2}\).

Step 5: Expressing the Answer in Reduced Form

The expression \(\frac{13}{12x^2}\) is already in its simplest form because \(13\) and \(12\) share no common divisors besides \(1\). Thus, the expression is fully simplified.

Key Concepts

Least Common Denominator (LCD)

When adding or subtracting fractions, identifying the Least Common Denominator (LCD) is crucial because it allows us to combine fractions by rewriting them with the same denominator.

The LCD is essentially the smallest multiple that two or more denominators share. It can be found by determining the lowest common multiple of the fraction's denominators.

In our exercise, we were working with fractions \(-5/(4x^2)\) and \(7/(3x^2)\). With denominators \(4x^2\) and \(3x^2\):

• First, identify the numerical part: The least common multiple (LCM) of 4 and 3 is 12, which provides the numerical part of our common denominator.
• Second, include the variable part: Since both denominators involve \(x^2\), the LCD incorporates this as well, making it \(12x^2\).

Thus, \(12x^2\) ensures the fractions can be expressed with the same denominator, facilitating addition.

Simplifying Fractions

Simplifying fractions is about rewriting them in their simplest form without changing their value. This involves ensuring that the numerator and the denominator share no common factors other than 1.

To simplify, follow these steps:

• Factor both numerator and denominator into their prime components.
• Cancel out any common factors.

For example, when we achieved our final result \(\frac{13}{12x^2}\), we needed to verify its simplicity. \(13\) is a prime number and shares no common factors with \(12\) other than 1, indicating that the fraction is already simplified.

This ensures that the expression is in its most reduced form, making it easier to interpret and work with in further calculations.

Adding Fractions

Adding fractions involves combining their numerators after adjusting them to have the same denominator. This process ensures accurate addition and simplifies handling fractions in algebra.

The steps involve:

• Equalizing the denominators using the LCD.
• Rewriting each fraction equivalent to maintain their original values, by multiplying both numerator and denominator by the necessary factors.
• With the same denominators, directly add the numerators together.

For instance, by converting \(\frac{-5}{4x^2}\) and \(\frac{7}{3x^2}\) to \(\frac{-15}{12x^2}\) and \(\frac{28}{12x^2}\) respectively, they share a common denominator \(12x^2\). The sum becomes: \[\frac{-15+28}{12x^2} = \frac{13}{12x^2}\] This method ensures each fraction remains true to its original value while allowing them to be easily combined.