Problem 19 - Algebra and Trigonometry

Question

\(15-20\) : 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 simplifies to \\(-5x + 10y\\).

Step 1: Distribute the Negative Sign and the Fraction

To eliminate the parentheses, we distribute \(-\frac{5}{2}\) through the expression inside the parentheses \(2x - 4y\). This means multiplying \(-\frac{5}{2}\) by each term inside: - \[ \left(-\frac{5}{2}\right) \cdot 2x = -\frac{5}{2} \times 2x = -5x \] - \[ \left(-\frac{5}{2}\right) \cdot (-4y) = -\frac{5}{2} \times (-4y) = 10y \]

Step 2: Write Final Expression

Now that we have distributed \(-\frac{5}{2}\) to each term inside the parentheses, we have the expression without any parentheses: \(-5x + 10y\)

Key Concepts

Properties of Real Numbers

When dealing with algebraic expressions, understanding the properties of real numbers is essential in simplifying and manipulating equations. These properties include important rules such as the distributive, commutative, and associative properties, which help us to rearrange and simplify expressions efficiently.
The **distributive property** is particularly useful when removing parentheses in an expression. It states that for any three real numbers, say a, b, and c,

• \( a(b + c) = ab + ac \)

In the original problem, we saw this property in action as we distributed \(-\frac{5}{2}\) to both terms inside the parentheses, which were \(2x\) and \(-4y\). By applying this rule correctly, we were able to eliminate the parentheses and simplify the expression into \(-5x + 10y\).
Additionally, understanding other properties like the **commutative property**, which allows changing the order of terms without changing the result, can prove useful. Similarly, the **associative property** which refers to regrouping terms can assist in managing more complex expressions. However, for the operations in this exercise, the distributive property was the main tool.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation. In the exercise, the expression within the parentheses, \(2x - 4y\), is an example of an algebraic expression.
These expressions are often manipulated to solve equations, find values of variables, or simplify complex expressions. They are foundational in algebra because they succinctly represent mathematical relationships and allow us to work with unknown quantities.
Understanding the components of algebraic expressions is important:

• **Variables** - letters like \(x\) and \(y\) that represent unknown values.
• **Coefficients** - numbers that are multiplied by the variables, such as \(2\) and \(-4\) in the expression.
• **Terms** - parts of the expression which are separated by addition or subtraction, \(2x\) and \(-4y\) are terms.

These components can be manipulated using the properties of real numbers to achieve a desired form or solution, as seen in the exercise.

Simplifying Expressions

Simplifying expressions involves combining like terms and applying properties of real numbers to make expressions more manageable. In the provided exercise, we simplified \(-\frac{5}{2}(2x - 4y)\) by distributing and removing parentheses. Simplifying expressions allows us to express mathematical relationships in the simplest form.
When simplifying, always look for opportunities to:

• **Distribute** to eliminate parentheses.
• **Combine like terms** (terms with the same variables raised to the same power). For example, if the expression was longer with more terms like \(-5x + 3x\), those could be combined to form \(-2x\).
• **Convert fractions or decimals** when necessary to make calculations easier or clearer.

Always check your work to ensure no mathematical rules are violated. Simplification aims to produce the clearest and most straightforward form of the expression, making it easier to solve equations or apply further mathematics.