Problem 90 - Fundamentals of Mathematics

Question

For problems \(57-140\), solve each equation. $$ \frac{3 y}{2}-4=6 $$

Step-by-Step Solution

Answer

The solution is \( y = \frac{20}{3} \).

Step 1: Isolate the Fraction Term

Start by isolating the term with the fraction. Add 4 to both sides to move the constant term to the right side of the equation: \[ \frac{3y}{2} - 4 + 4 = 6 + 4 \] The equation simplifies to: \[ \frac{3y}{2} = 10 \]

Step 2: Eliminate the Fraction

To eliminate the fraction, multiply both sides of the equation by 2, which is the denominator of the fraction: \[ 2 \times \frac{3y}{2} = 10 \times 2 \] This simplifies to: \[ 3y = 20 \]

Step 3: Solve for y

Now that the equation is simplified and the fraction is gone, divide both sides by 3 to solve for \( y \): \[ y = \frac{20}{3} \]

Key Concepts

Fraction Elimination

When working with an equation that includes a fraction, it is often best to eliminate the fraction. This makes the equation easier to handle. Fractions can be confusing, so removing them is helpful for clear equations. To eliminate a fraction in an equation, you should:

• Identify the denominator (the bottom number of the fraction).
• Multiply every term on both sides of the equation by that denominator.

By multiplying every term by the denominator, you turn the fraction into a whole number. For example, in the equation \( \frac{3y}{2} = 10 \), multiplying both sides by 2 removes the fraction, turning it into \( 3y = 20 \). This process helps to simplify the equation, making the next steps easier.

Isolation of Variables

In algebra, isolating the variable is a key step to solving equations. Isolation means we want to get the variable alone on one side of the equation. This helps us easily find its value. Here's how you can isolate a variable:

• Use addition or subtraction to move constant terms to the opposite side of the equation.
• If the variable is part of a fraction, first eliminate the fraction by multiplying by the denominator, as discussed before.

When you have a term like \( \frac{3y}{2} - 4 = 6 \), you start by removing any numbers with addition or subtraction. Add 4 to both sides to isolate the fraction term, resulting in \( \frac{3y}{2} = 10 \). Isolating the variable makes it simpler to solve the rest of the equation.

Equation Simplification

Simplifying equations is about making them easier to solve. It often involves removing fractions, like in the previous steps, and simplifying mathematical expressions. Here’s what simplification generally involves:

• Perform basic arithmetic to reduce complex expressions when possible.
• Combine like terms.
• Ensure the equation is as straightforward as possible for solving.

Consider the equation \( 3y = 20 \) from the earlier steps. Dividing both sides by 3 simplifies the equation to \( y = \frac{20}{3} \). This is a simplified solution. The goal here is to have a clear and final answer, which makes it easy to understand the value of the variable. Simplification tidies up your work and ensures accuracy in results.