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Chapter 3: Problem 28

Simplify the expression. $$2(10-6 x)$$

Short Answer

Expert verified
The simplified expression is \(20 - 12x\).

Step by step solution

01

Apply the Distributive Property

To simplify the expression \(2(10-6x)\), we start by applying the distributive property. This means we multiply the number outside the parenthesis by each term inside the parenthesis. Thus, multiply \(2\) by \(10\) and also by \(-6x\).
02

Perform the Multiplication

Calculate each of the multiplications from the previous step separately:1. \(2 \times 10 = 20\)2. \(2 \times (-6x) = -12x\)
03

Combine the Terms

Combine the results from the multiplications into a single expression. After performing the multiplications, the expression becomes:\[ 20 - 12x \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Expression Simplification
Simplifying an expression involves making it as easy to understand as possible by removing any unnecessary parts. When you see an expression like \(2(10-6x)\), it's not in its simplest form because the terms are bundled inside parentheses. The goal here is to rewrite it without parentheses.
Start by looking for opportunities to distribute, multiply, and then simplify. By carrying out these steps, you end up with a straightforward and clearer expression. This process is key in algebraic manipulations, as it helps in solving equations efficiently.
Multiplication in Algebra
In algebra, multiplication is used often to simplify and solve expressions. When you encounter an expression like \(2(10-6x)\), the distributive property is your friend and will guide you in multiplying each term inside the parentheses by the term outside.
Here's how it works:
  • Multiply \(2\) by \(10\) to get \(20\).
  • Next, multiply \(2\) by \(-6x\) to get \(-12x\).
Take each result and recombine them to form a neat expression: \(20 - 12x\). This showcases the critical role of multiplication in transforming and simplifying algebraic expressions.
Combining Like Terms
Once you have performed multiplication using the distributive property, the next step is to combine like terms to further simplify the expression. Like terms are terms that have the same variable raised to the same power. They can be added or subtracted from one another easily.
In our example, \(20 - 12x\), there are no like terms to combine since \(20\) is a constant and \(-12x\) is a term with a variable. Therefore, the expression is already simplified at this stage.
Knowing how to identify and combine like terms is crucial because it reduces complexity and prepares the expression for solving equations or substituting values.

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