Problem 30 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$-10(b-1)+4 b$$

Step-by-Step Solution

Answer

The simplified form of the expression \(-10(b-1)+4b\) is \(-6b+10\).

Step 1: Apply distributive property

First, distribute the \(-10\) across the expression \((b-1)\), resulting in \(-10b+10\).

Step 2: Combine with the term \(4b\)

Then, combine \(-10b+10\) and \(4b\), resulting in \(-10b+10+4b\) or, more simply, \(-10b+4b+10\).

Step 3: Combine like terms

Finally, combine the like terms (\(-10b\) and \(4b\)) which results in \(-6b+10\). This is the simplest form of the given expression.

Key Concepts

Distributive Property

The distributive property is a fundamental concept in simplifying algebraic expressions. It allows you to break down expressions into simpler parts by distributing a multiplier across terms inside parentheses. When you apply the distributive property, you essentially multiply each term inside the parentheses by the factor outside it. This is written as follows:

• For any numbers or expressions, if you have a term like \( a(b+c) \), you use the distributive property to write it as \( ab + ac \).

In our specific example, the expression \(-10(b-1)\) requires you to distribute the \(-10\) to both \(b\) and \(-1\). Thus, you multiply \(-10\) by \(b\) to get \(-10b\), and \(-10\) by \(-1\), which results in \(+10\).
This process transforms the original expression into \(-10b + 10\). Remember, the distributive property is valuable because it helps eliminate parentheses, paving the way to simplify further by combining like terms.

Combining Like Terms

Once the distributive property has been applied, the next step is to combine like terms. This is a critical aspect of simplifying algebraic expressions. Like terms are terms that have identical variable parts and powers. You can only combine them by performing arithmetic operations on their coefficients.

For instance:

• In the expression \(-10b + 4b + 10\), both \(-10b\) and \(4b\) are like terms because they share the same variable "b" with the same power, which is 1. The \(10\) is a constant term and does not combine with the variable terms.

To simplify, you add the coefficients of the \(b\) terms: \(-10 + 4 = -6\). The constant term \(+10\) remains as is. Therefore, the expression becomes \(-6b + 10\).
This step effectively reduces the complexity of the expression by consolidating terms, which makes any expression much simpler to read and work with.

Algebraic Expressions

Understanding algebraic expressions is crucial for simplifying them effectively. An algebraic expression is a combination of numbers, variables, and arithmetic operations (such as addition, subtraction, multiplication, and division).

Here's what you need to know about them:

• They can include constants (like \(10\)), variables (like \(b\)), and coefficients (like \(-10\) in \(-10b\)).
• Algebraic expressions do not have an equality sign, unlike equations.
• Simplifying them often involves utilizing the distributive property and combining like terms, as we did in our example.

In this specific case, the expression \(-10(b-1) + 4b\) is made up of numbers and the variable \(b\). Simplifying such expressions is essential, as it allows us to understand and manipulate them more easily. Once simplified, they are less cumbersome and more straightforward to integrate into further calculations or problem-solving scenarios.