Problem 36 - Introductory Algebra for College Students

Question

Solve each rational equation. $$\frac{10}{y+2}=3-\frac{5 y}{y+2}$$

Step-by-Step Solution

Answer

The solution to the rational equation is \(y = -2\). Check the solution by substituting \(-2\) for \(y\) in the original equation. Please note that any solution that makes the denominator zero would be excluded, because division by zero is undefined.

Step 1: Identify the LCD

Observe the equations and note that the fractions have the same denominator of \(y+2\). So the least common denominator (LCD) is \(y+2\)

Step 2: Remove fractions

Multiply each side of the equation by the LCD, which is \(y+2\). This process will remove fractions. The equation becomes \(10 = 3(y+2) - 5y\)

Step 3: Simplify the equation

Simplify the equation by distributing 3 in the right side of the equation, which gives: \(10 = 3y + 6 - 5y\). Then combine like terms, and we get \(10 = -2y + 6\)

Step 4: Solve for \(y\)

Rearrange the equation by moving \(6\) to the left side of the equation, and we get \(4 = -2y\). Divide both sides by \(-2\), we get \(y = -2\)

Key Concepts

Least Common Denominator

In solving rational equations like the one given, finding the least common denominator (LCD) is a crucial step. The LCD is the smallest number that all denominators in the equation can divide into evenly. It's akin to finding a common ground where all fractions can 'meet' to be combined or compared. For instance, consider the denominators in our example, which are both \(y+2\). Since they are the same, the LCD is simply \(y+2\). This convenient sameness allows us to consolidate the equation more straightforwardly.

When the denominators are different, we look for the smallest expression that each denominator can go into without leaving a remainder. This might involve factoring the denominators and finding a product that includes each factor at least once. In practice, this step simplifies the work by transforming an equation with multiple fractions into a form where you can easily perform algebraic operations like addition, subtraction, multiplication, and division without dealing with fractions at each turn.

Algebraic Fraction Simplification

Moving beyond the LCD, the process of algebraic fraction simplification is essential in managing more complex rational expressions. Simplification might involve reducing fractions to their simplest form by cancelling common factors in the numerator and denominator, or combining fractions with a common denominator.

In the given exercise, the process starts by multiplying each term by the LCD to eliminate the fractions. Simplification continues by expanding any products and combining like terms, making the equation easier to solve. Remember, the goal of simplification is to make the equation as straightforward as possible. This might sometimes require us to factor expressions further or divide both sides of the equation by a common factor, guiding us one step closer to the solution.

Linear Equations in Algebra

The term \(y\) in our exercise is not raised to any power higher than one; therefore, the equation can be classified as a linear equation. These equations form the basis for much of algebra and are characterized by each term being either a constant or the product of a constant and a single variable. Solving linear equations usually involves isolating the variable on one side of the equation.

To solve linear equations like the one in our example, you usually follow a sequence of steps: distribute any multiplied values across parentheses, combine like terms, isolate the variable term, and divide through by the variable's coefficient to solve for the variable. The solution to the exercise, where \(y=-2\), is the result of simplifying and rearranging the equation following these steps, demonstrating the systematic approach to solving linear equations in algebra.

College-Level Mathematics

Solving rational equations is a vital skill in college-level mathematics, serving as a foundation for more advanced topics in calculus, engineering, physics, economics, and beyond. Proficiency in this area requires not only understanding specific procedures, like finding the LCD or simplifying fractions, but also recognizing the underlying principles of algebra that apply to a wide range of mathematical problems.

College-level math typically involves abstract thinking and the ability to generalize concepts to solve a variety of problems. For example, the techniques used to solve the given rational equation are not limited to only this type of problem but are also applicable to any situation where simplification and manipulation of algebraic expressions are required. Strengthening these skills through practice allows students to tackle complex scenarios with confidence and precision.