91Ó°ÊÓ

Fractions are an essential concept in mathematics, representing a part of a whole. They are composed of two parts: the numerator, which is the top number, and the denominator, the bottom number. The numerator indicates how many parts you have, while the denominator shows into how many parts the whole is divided. For example, in the fraction \( \frac{1}{2} \), 1 is the numerator, and 2 is the denominator, indicating one part out of two equal parts of a whole.

When dealing with fractions in equations, it is important to understand how to add, subtract, multiply, and divide them. Adding or subtracting fractions requires a common denominator, which leads us to the next core concept: the least common denominator.

The least common denominator (LCD) is the smallest number that can serve as a common denominator for a set of fractions. Finding the LCD is essential for adding or subtracting fractions because having a common denominator enables us to perform these operations more easily.

In the context of rational equations like \( \frac{1}{2} + \frac{2}{y} = \frac{1}{3} + \frac{3}{y} \), we use the LCD to eliminate fractions altogether. This makes the problem easier to solve. Here, we determined the LCD to be \(6y\), which is the smallest number that both 2, 3, and \(y\) will divide evenly. By multiplying each term of the equation by the LCD, we eliminate the fractions and simplify the equation.

Simplification is the process of reducing an expression or an equation to its simplest form. When we simplify fractions, we reduce them to the smallest possible numerator and denominator that retain the same value. For example, \( \frac{6}{12} \) simplifies to \( \frac{1}{2} \) because both the numerator and the denominator can be divided by 6.

In equation solving, simplification involves distributing products and combining like terms to make an equation more manageable. After eliminating the fractions using the LCD, our rational equation \( 6y(\frac{1}{2} + \frac{2}{y}) = 6y(\frac{1}{3} + \frac{3}{y}) \) was simplified to \( 3y + 12 = 2y + 18 \). This simplification step makes it easier to solve for the variable \(y\).

Equation solving is the process of finding the value of the variable that satisfies the equation. It involves performing a series of algebraic operations to isolate the variable. After simplifying the rational equation to \(3y + 12 = 2y + 18\), our task was to solve for \(y\).

We did this by isolating \(y\), which involved subtracting \(2y\) from both sides, resulting in \(y + 12 = 18\). Further subtraction of 12 from both sides gave us \(y = 6\). This value was then substituted back into the original equation to verify its correctness: \( \frac{1}{2} + \frac{1}{3} = \frac{1}{3} + \frac{1}{2} \), both simplifying to \( \frac{5}{6} \). Verification tells us that our solution, \(y = 6\), is indeed correct.