Problem 16 - Precalculus

Question

Solve each equation. $$\frac{1}{y}+1=\frac{3}{y}-\frac{1}{2 y}$$

Step-by-Step Solution

Answer

The solution is \( y = \frac{3}{2} \).

Step 1: Find a Common Denominator

To solve the equation \( \frac{1}{y} + 1 = \frac{3}{y} - \frac{1}{2y} \), start by finding a common denominator for the fractional terms. Here, it is clear that \( 2y \) can serve as the common denominator since 2 is the smallest multiple of 1 and 2.

Step 2: Rewrite the Terms

Rewrite each term in the equation with the denominator \( 2y \). This gives: \( \frac{2}{2y} + \frac{2y}{2y} = \frac{6}{2y} - \frac{1}{2y} \).

Step 3: Simplify the Equation

Combine the fractional terms on both sides of the equation: \( \frac{2 + 2y}{2y} = \frac{6 - 1}{2y} \), simplifying to \( \frac{2 + 2y}{2y} = \frac{5}{2y} \).

Step 4: Clear the Fractions

Since both sides of the equation have \( 2y \) as the denominator, multiply both sides by \( 2y \). This results in: \( 2 + 2y = 5 \).

Step 5: Solve for y

Isolate \( y \) by subtracting 2 from both sides: \( 2y = 5 - 2 \), which simplifies to \( 2y = 3 \). Divide both sides by 2 to solve for \( y \), yielding \( y = \frac{3}{2} \).

Key Concepts

Common Denominator

When working with rational equations, one of the initial steps is ensuring terms have a common denominator. This process simplifies the equation, making it easier to manipulate and solve. Consider fractions like \( \frac{1}{y} \) and \( \frac{3}{y} \). Both share the same denominator, \( y \), which is straightforward. However, adding another fraction like \( \frac{1}{2y} \) requires a more uniform base.
The goal is to transform all fractions so they share one common denominator. In our example, \( 2y \) serves well, as it's a multiple of both \( y \) and \( 2y \). Achieving a common denominator allows us to combine fractions by aligning the denominators through equivalent fractions.

Fraction Simplification

Once you've established a common denominator for your rational equation, the next step is simplification. This involves converting each fraction with the newly chosen common denominator. Take the example: \( \frac{2}{2y} \) and \( \frac{6}{2y} \). Simplification involves these sub-tasks:

• Ensure that each term is rewritten so the denominator is \( 2y \).
• Adjust numerators accordingly, preserving the equation's equality.

In our problem, each term was rewritten, and then their numerators adjusted. For instance, \( 1 \) became \( \frac{2y}{2y} \), making it preservable in the equation. Simplifying fractions allows for straightforward consolidation of terms next.

Solving for Variables

With an equation reduced to a single common denominator, the task becomes solving for the variable. Start by eliminating the denominators—multiply both sides accordingly to clear them. With the equation \( \frac{2 + 2y}{2y} = \frac{5}{2y} \), focus on reducing its complexity:
1. Multiply both sides by \( 2y \) to clear denominators.
2. This operation leaves you with the simplified equation \( 2 + 2y = 5 \).
Now, isolate the variable \( y \) by rearranging terms. From \( 2y = 3 \), divide both sides by 2 to find that \( y = \frac{3}{2} \). Isolating variables includes rearranging and, most importantly, solving for a concrete value.

Equation Balance

Balancing an equation is crucial for maintaining equality between both sides throughout manipulations. Essentially, whatever changes you apply to one side must also affect the other. While solving \( 2 + 2y = 5 \), notice the systemic balance with each step:

• Subtract 2 from both sides, maintaining equilibrium.
• Solve the resulting \( 2y = 3 \) by dividing by 2, remaining balanced.

Overlooking balance can lead to incorrect solutions and disrupts solving the equation process. Always check your operations to preserve this fundamental principle. Balancing equations ensures each side remains equal, fostering accurate problem-solving.