Problem 124 - Elementary and Intermediate Algebra

Question

Simplify each expression, if possible. $$ -(z+2)+5(3-z) $$

Step-by-Step Solution

Answer

The simplified expression is \(-6z + 13\).

Step 1: Distribute Negative Sign

First, simplify the expression by distributing the negative sign across the terms inside the parenthesis \( -(z+2) \). This gives: \(-z - 2\).

Step 2: Distribute the Number 5

Next, distribute the 5 into the expression \( (3-z) \), which results in \( 5 \times 3 - 5 \times z = 15 - 5z \).

Step 3: Combine Like Terms

Combine the simple expressions resulting from the distribution: \(-z - 2 + 15 - 5z\). Combine the \(-z\) and \(-5z\) terms, and combine the constants \(-2\) and \(15\). This simplifies to \(-6z + 13\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows you to multiply a single term across the terms within a parenthesis. This property is valuable when simplifying expressions because it helps break down complex formulas into simpler parts.
For the exercise given, the distributive property was applied twice.

• First, when distributing the negative sign across the terms inside the parentheses: \[-(z+2) = -z - 2\]
• Second, when distributing the number 5 across the terms: \[5(3-z) = 15 - 5z\]

The distributive property ensures that each term inside the parentheses is multiplied by the factor outside, facilitating the simplification of the expression. Essentially, it ensures that all components of an expression are accounted for in your calculations.

Combining Like Terms

Combining like terms is crucial in simplifying algebraic expressions. This concept involves merging terms that have the same variable component, ensuring that your final expression is as compact as possible.
In the exercise:

• We had terms \(-z\) and \(-5z\), which combine to \(-6z\). These are like terms since they both have the variable \(z\).
• Similarly, the constants \(-2\) and \(15\) were combined to form \(13\).

The process of combining like terms involves identifying terms with the same variables and simply adding or subtracting their coefficients. This reduces the complexity of expressions and brings clarity to your calculations.

Negative Numbers

Understanding how to handle negative numbers is essential when simplifying expressions. In algebra, negative signs affect the direction of operations. When distributing a negative sign, it reverses the sign of each term inside the parentheses.
In the exercise, distributing the negative sign changed the terms inside the parenthesis from \(z+2\) to \(-z-2\). Similarly, when we combined numerical constants, like \(-2\) with \(15\), we needed to take care to maintain the correct sign, resulting in \(13\).
The key to working with negative numbers is to remember:

• Adding a negative number is the same as subtracting.
• Subtracting a negative number is the same as adding, because it reverses the sign.
• Multiplying or dividing two negative numbers yields a positive result.

By keeping these rules in mind, we can confidently and accurately simplify expressions that involve negative numbers.