Problem 27 - Algebra and Trigonometry

Question

Use properties of real numbers to write the expression without parentheses. \(-\frac{5}{2}(2 x-4 y)\)

Step-by-Step Solution

Answer

The expression without parentheses is \(-5x + 10y\).

Step 1: Identify the Expression

We start by identifying the given expression: \(-\frac{5}{2}(2x - 4y)\). Our task is to remove the parentheses by applying properties of real numbers.

Step 2: Apply the Distributive Property

We use the distributive property, which states that for any real numbers \(a\), \(b\), and \(c\), \(a(b + c) = ab + ac\), and taking care of signs:\(-\frac{5}{2}(2x - 4y) = -\frac{5}{2} \times 2x - (-\frac{5}{2} \times 4y)\).

Step 3: Simplify Each Term

First, simplify each part:- \(-\frac{5}{2} \times 2x = -5x\)- \(-\frac{5}{2} \times (-4y) = 10y\). This simplification uses basic multiplication of fractions and numbers, noting that two negatives multiply to make a positive.

Step 4: Write the Final Expression

Combine the simplified terms to form the expression without parentheses: \(-5x + 10y\).

Key Concepts

Understanding Real Numbers

Real numbers include all the numbers on the number line. This means both rational numbers, such as integers, fractions, and decimals, as well as irrational numbers like √2 and π that cannot be expressed as a simple fraction.
Real numbers are very useful because they allow us to describe a vast range of quantities and make up the foundation of most of mathematics.
In the context of the exercise, we deal with numbers like \(-\frac{5}{2}\), 2, \(4y\), and more, which are all part of this comprehensive set.

• Rational Numbers: Numbers that can be expressed as a fraction, such as \(\frac{1}{2}\), -3, or 5.
• Irrational Numbers: Numbers that cannot be precisely written as a simple fraction, including numbers like \(\pi\) and \(\sqrt{2}\).
• Using the Distributive Property: Allows us to manipulate and simplify expressions involving real numbers.

Understanding real numbers lays the groundwork for mastering operations like addition, subtraction, and importantly, distribution over an expression as seen in the exercise.

Mastering Simplification

Simplification is the process of making a mathematical expression as simple as possible. This means performing operations collaboratively to reduce expressions to fewer terms or a more familiar form.
In the given exercise, simplification helps make the problem more manageable. After applying the distributive property, you simplify parts of the equation as shown:- \(-\frac{5}{2} \times 2x = -5x\)- \(-\frac{5}{2} \times (-4y) = 10y\)

• Multiply fractions and whole numbers carefully: The interaction between negative signs must be correctly handled to avoid errors.
• Combine like terms: Once simplified, you rewrite the expression with as few terms as possible.

By simplifying, we transform the original, perhaps cumbersome expression to a cleaner "-5x + 10y" expression, making it easier to understand and use.

Expressions Without Parentheses

Expressions without parentheses are often easier to work with because they are in their simplest form. The removal of parentheses in a mathematical expression can be achieved through the distributive property.
This guarantees all terms inside the parentheses are correctly influenced by the number outside.
Let's break it down using our example:1. **Start with**: \(-\frac{5}{2}(2x - 4y)\)2. **Apply Distribution**: Every item inside the parentheses gets multiplied by \(-\frac{5}{2}\).3. **Simplified Results**: - \(-\frac{5}{2} \times 2x = -5x\) - \(-\frac{5}{2} \times (-4y) = 10y\)

• Checking Results: Always verify each step during distribution to ensure accuracy.
• Benefits: Without parentheses, expressions become straightforward for further algebraic operations.

This method not only simplifies the problem but also unveils a clearer expression like "-5x + 10y" without any enclosing brackets.