Problem 86 - Prealgebra and Introductory Algebra

Question

Rewrite the expression as the sum of two fractions in simplest form. $$\frac{6 x+7 y}{x y}$$

Step-by-Step Solution

Answer

The sum of two fractions in simplest form for the given expression is \( \frac{6}{y} + \frac{7}{x}\)

Step 1: Break down the fraction into two fractions

The expression can be split into two fractions as follows: \(\frac{6x}{xy} + \frac{7y}{xy}\)

Step 2: Simplify the fractions

The fractions can be simplified by reducing terms. The first fraction simplifies to \(\frac{6}{y}\) and the second fraction simplifies to \(\frac{7}{x}\)

Step 3: Present the final answer

The original expression \(\frac{6x + 7y}{xy}\) when rewritten as the sum of two fractions in simplest form is \(\frac{6}{y} + \frac{7}{x}\)

Key Concepts

Breaking Down Fractions

When faced with a complex algebraic fraction, such as \( \frac{6x + 7y}{xy} \), the goal is to break down the fraction into simpler parts. This process involves separating the original fraction into individual terms that together add up to the original fraction. It's like taking a puzzle apart to see each piece clearly.

For our given expression, the numerator consists of two terms, 6x and 7y, which are being divided by xy. The key step is to split this into two separate fractions: \( \frac{6x}{xy} \) and \( \frac{7y}{xy} \). This allows us to focus on simplifying each term independently, making it easier to see and reduce any common factors between the numerator and the denominator.

Reducing Algebraic Expressions

Reducing algebraic expressions is a fundamental skill in simplifying mathematical problems. The aim is to modify the expression to its most basic form without changing its value. After breaking down the larger fraction into \( \frac{6x}{xy} \) and \( \frac{7y}{xy} \) as in our example, you can reduce these fractions by canceling out common factors in both the numerator and the denominator.

For \( \frac{6x}{xy} \), x is a common factor in the numerator and denominator and thus can be cancelled out, leaving us with \( \frac{6}{y} \). Similarly, for \( \frac{7y}{xy} \), y can be cancelled out, resulting in \( \frac{7}{x} \). This process not only simplifies the fractions but also makes the overall expression more comprehensible and easier to work with in subsequent calculations.

Sum of Fractions in Simplest Form

The sum of fractions refers to the result of adding two or more fractions together. To add fractions effectively, they must be expressed in simplest form; this means they have been reduced as much as possible, as we did in the previous steps with \( \frac{6}{y} \) and \( \frac{7}{x}\).

When the fractions in question share a common denominator, they can simply be added together by combining their numerators. In the case of our example, we do not have a common denominator. After simplification, we are left with the expression \( \frac{6}{y} + \frac{7}{x} \) which represents the original fraction written as the sum of two simpler fractions. This final form is valuable because it's much more straightforward and allows for easier manipulation and understanding in further operations or applications, such as integration or finding common denominators.