Problem 35 - Elementary and Intermediate Algebra

Question

Multiply, and then simplify, if possible. \(12 y\left(\frac{5 y-8}{6 y}\right)\)

Step-by-Step Solution

Answer

The simplified expression is \(10y - 16\).

Step 1: Distribute the 12y

To multiply the expression, start by distributing the term outside the parentheses, which is \(12y\), to each term inside the parentheses. Thus, multiply \(12y\) by each term of \( \frac{5y-8}{6y}\).

Step 2: Multiply each term separately

First, multiply \(12y\) by \(\frac{5y}{6y}\): \[12y \times \frac{5y}{6y} = \frac{12y \times 5y}{6y}\]Second, multiply \(12y\) by \(\frac{-8}{6y}\):\[12y \times \frac{-8}{6y} = \frac{12y \times -8}{6y}\]

Step 3: Simplify the products

In the first term, notice that \(y\) in the numerator and denominator cancels out:\[\frac{12y \times 5y}{6y} = \frac{12 \times 5y}{6} = 10y\]In the second term:\[\frac{12y \times -8}{6y} = \frac{12 \times -8}{6} = -16\]Remember that \(y\) cancels out again.

Step 4: Combine the simplified terms

Add together the simplified terms from Step 3:\[10y - 16\]This is your final answer.

Key Concepts

Understanding the Distributive Property

The distributive property is an essential skill in algebra, helping you to simplify complex expressions. When you see a term outside parentheses, like in the exercise with 12y, you can "distribute" that term to each part within the parentheses. Imagine unpacking a box that contains smaller boxes; you need to touch each item to unpack everything. That’s what the distributive property does mathematically.

Mathematically, it’s expressed as:

• For any numbers or expressions a, b, and c:
  \( a(b + c) = ab + ac \)

This principle allows you to transform and work with algebraic expressions more flexibly. By distributing 12y to each fraction segment in the exercise, you break down the task into simpler components that are easier to manage.

Simplifying Expressions

Simplifying expressions in algebra is like tidying up a messy room. By organizing terms and reducing fractions, you create a cleaner, more manageable version of the expression.In the provided exercise, the initial step involved distributing 12y across the given fractions. Simplifying started when each product was separately considered. Observing factors in numerators and denominators allows for cancellation, shrinking terms down to their simplest form. For example:

• In \( \frac{12y \times 5y}{6y} \), the \( y \) in the numerator cancels with the \( y \) in the denominator. This leaves \( \frac{12 \times 5y}{6} = 10y \) upon simplification.
• For \( \frac{12y \times -8}{6y} \), \( y \) cancels out as well, leading to \( \frac{12 \times -8}{6} = -16 \).

Simplifying expressions is not just about canceling terms; it's about applying operations consistently to achieve a neat and understandable form like 10y - 16 in the exercise.

Mastering the Art of Multiplying Fractions

Multiplying fractions might seem a bit tricky at first, but it's straightforward once you know the rule. Simply multiply the numerators together and the denominators together. This basic concept is used repeatedly in algebra, as well as in everyday math tasks.From our exercise, let's break it down:

• Multiply the numerators: For \( 12y \times \frac{5y}{6y} \), the top becomes \( 12y \times 5y \), and for \( 12y \times \frac{-8}{6y} \), it becomes \( 12y \times -8 \).
• Multiply the denominators: Both expressions share the \( 6y \) denominator. Multiplying the denominators ensures the fractions are consistent in form.

Understanding the multiplication of fractions simplifies managing expressions. It allows for systematic reduction and easier computation, as seen with the simplification to 10y - 16.