Problem 23 - Precalculus Mathematics for Calculus

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

Step 1: Distribute the First Term

Apply the distributive property to remove the parentheses. Multiply \(-\frac{5}{2}\) by each term inside the parentheses. We have:\[ -\frac{5}{2} \times (2x - 4y) = -\frac{5}{2} \times 2x + \left(-\frac{5}{2} \right)\times (-4y) \] This simplifies to:\[ -5x + 10y \]

Step 2: Simplify Each Term

The expression after distribution is \(-5x + 10y\). Each term is already in its simplest form, so simplifying further just involves ensuring each term is as reduced as possible.Thus, the expression without parentheses and fully simplified is:\[ -5x + 10y \]

Key Concepts

Properties of Real Numbers

When dealing with algebraic expressions like \( -\frac{5}{2}(2x - 4y) \), it is essential to understand the Properties of Real Numbers. These properties help us manipulate and transform expressions into simpler forms. One of the most important of these properties is the **Distributive Property**:

• **Distributive Property**: This property states that for any real numbers \( a, b, \) and \( c \), \( a(b + c) = ab + ac \). It allows us to distribute a factor across terms inside parentheses.

In our given expression, we used the distributive property to multiply \( -\frac{5}{2} \) with each term inside the parentheses: \( 2x \) and \( -4y \).
This property turns complex algebraic expressions into simple and manageable ones. Knowing when and how to apply it is crucial in algebra.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form. This process makes the expression easier to understand and work with. Here, we need to simplify each term of the expression \( -5x + 10y \) after using the distributive property.
The terms \(-5x\) and \(10y\) are both products of constants and variables. Since there are no like terms, we simply ensure that the coefficients and variables are as reduced as possible.

• **Coefficients**: Numerical factors should be simplified, in this case, \(-5\) and \(10\) are already as simple as they can be.
• **Variables**: Ensure there are no similar terms that can be combined. Here, \(x\) and \(y\) are different variables and thus remain separate.

Simplifying doesn't change the value, it just helps us express the idea more clearly. It’s a vital skill in algebra.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. The expression \( -\frac{5}{2}(2x - 4y) \) is an example of such a phrase. Understanding how to work with algebraic expressions is fundamental in algebra.
Key components of algebraic expressions include:

• **Constants**: Numbers on their own like \(-\frac{5}{2}\).
• **Variables**: Symbols that stand for numbers, such as \(x\) and \(y\).
• **Operators**: Symbols that indicate operations like addition or subtraction.
• **Terms**: Parts of the expression separated by operators, such as \(-5x\) and \(10y\) after distribution.

Algebraic expressions like this one are used to represent relationships and solve problems in algebra. Mastering how to manipulate and simplify them is a stepping stone to mastering algebra itself.