Problem 21 - Fundamentals of Mathematics

Question

Simplify each expression by combining like terms. $$(-5+3) a-(2+5) b-(3+8) b$$

Step-by-Step Solution

Answer

The simplified expression is \(-2a - 18b\).

Step 1: Simplify Parentheses

Start by simplifying the expressions inside the parentheses. For \(-5 + 3\), the result is \(-2\). For \(2 + 5\), the result is \(7\). And for \(3 + 8\), the result is \(11\). Rewrite the expression as \(-2a - 7b - 11b\).

Step 2: Combine Like Terms

Identify and combine like terms. The terms involving \(a\) and those involving \(b\) are like terms. In this expression, \(-2a\) is alone, and the \(b\) terms are \(-7b\) and \(-11b\). Combine \(-7b\) and \(-11b\) to get \(-18b\). So the expression becomes \(-2a - 18b\).

Key Concepts

Simplifying Expressions

In algebra, simplifying expressions is all about making expressions simpler without changing their value. This involves combining like terms and simplifying any parts of the expression, such as numbers inside parentheses or fractions. The goal is to rewrite the expression in the simplest form possible for easier calculation and understanding.

For example, consider the expression \((-5 + 3) a - (2 + 5) b - (3 + 8) b\). To simplify it, you first deal with any operations, such as subtraction or addition, inside the parentheses. Reducing these internal calculations makes it much easier to approach the overall expression.

By employing this technique, you convert the expression into a more transparent form, step by step, ensuring clarity and ease of further manipulation anytime you encounter a similar algebraic expression.

Combining Like Terms

Combining like terms is an essential process in algebra. Like terms are terms that have identical variable parts raised to the same power. For example, \(-7b\) and \(-11b\) have the same variable, \(b\), and can thus be combined.

Let's take the example given: after simplifying the parentheses, the expression becomes \(-2a - 7b - 11b\). Here, \(-7b\) and \(-11b\) are like terms. We can combine them to form \(-18b\). This simplifies the expression further to \(-2a - 18b\).

It's important to note that only like terms can be combined. Terms like \(-2a\) and \(-18b\) cannot be combined since their variable parts are different.

Parentheses Simplification

Parentheses simplification involves calculating and reducing any expression within parentheses first. Parentheses in an algebraic expression often indicate which operations should be completed first, guiding the order of operations in the expression.

Consider the expression \((-5 + 3) a - (2 + 5) b - (3 + 8) b\). By simplifying the terms in the parentheses, we get \(-2\), \(7\), and \(11\) respectively. This changes the original expression into \(-2a - 7b - 11b\).

Simplifying the parts within parentheses makes the rest of the expression easier to handle. It's crucial to perform these steps first to give a clear path for further simplification and combination of like terms.