Problem 31 - Introductory Algebra for College Students

Question

Solve each rational equation. $$\frac{8 y}{y+1}=4-\frac{8}{y+1}$$

Step-by-Step Solution

Answer

The solution to the equation is \( y = -1 \)

Step 1: Identifying the common denominator

In this problem, the common denominator across the entire equation is \( (y+1) \).

Step 2: Multiply through by the common denominator

Next, multiply each term of the equation by the common denominator \( (y+1) \). That simplifies the equation to:\n \( 8y = 4(y+1) - 8 \)

Step 3: Distribute and simplify

Next, distribute the 4 in the right side of the equation and simplify to get:\n \( 8y = 4y + 4 - 8 \n 8y = 4y - 4 \)

Step 4: Solve for y

Next, re-arrange the equation to isolate \( y \) on one side:\n \( 8y - 4y = - 4 \n 4y = -4 \n So, \( y = -4 / 4 = -1 \)

Key Concepts

Common Denominator

When solving rational equations, finding a common denominator is a crucial step. A rational equation is one that contains fractions with variables in the denominators. For example, consider the equation \( \frac{8y}{y+1} = 4 - \frac{8}{y+1} \). Both fractions here have the same denominator, which is \( (y+1) \).
This denominator allows us to simplify the problem by eliminating fractions, making it easier to solve the equation. Utilizing a common denominator is like speaking the same language for all fractions involved in the equation.

To handle it effectively:

• Identify a shared denominator among all fractions—here it is \( (y+1) \).
• Multiply every term in the equation by this common denominator, thereby canceling out the denominators.

This step helps to align terms, getting rid of fractions and transforming the problem into a simpler linear equation.

Equation Simplification

Once a common denominator is established and the equation is free from fractions, it's time to simplify the equation further. The goal here is to get the equation into a form that is easy to solve.

Consider the equation: \( 8y = 4(y+1) - 8 \).
To simplify:

• Distribute any numbers in front of parentheses—here, it's 4 on the right side, which gives \( 4y + 4 \).
• Combine like terms to condense the equation as much as possible—\( 4 - 8 \) simplifies to \( -4 \).

This leads to the simplified equation \( 8y = 4y - 4 \). Every simplification step reduces complexity, turning it into a straightforward algebraic expression and setting the stage for easily finding the variable.

Isolating Variables

The final and most important step is isolating the variable to find its value. This is done by manipulating the equation until the variable is alone on one side.

With the simplified equation \( 8y = 4y - 4 \), follow these steps:

• Re-order the terms to gather all terms containing the variable on one side. Subtract \( 4y \) from both sides to isolate terms with \( y \): \( 8y - 4y = -4 \).
• Perform any necessary arithmetic, such as division or subtraction, to further isolate \( y \). Here, divide both sides by 4 to get \( y = \frac{-4}{4} \).

This yields \( y = -1 \). Isolating the variable lets us solve for it directly, revealing the solution to the equation. This step requires careful repositioning of terms to neatly solve for the unknown.