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Chapter 3: Problem 27

Simplify the expression. $$2(10+\mathrm{x})$$

Short Answer

Expert verified
The simplified expression is \(20 + 2x\).

Step by step solution

01

Distribute the Outer Term

We start by distributing the coefficient \(2\) to both terms inside the parentheses \((10 + x)\). This means we multiply \(2\) by each term in the expression:\[2 \cdot 10 + 2 \cdot x\]
02

Simplify the Products

Now, calculate the products obtained from the distribution:- \(2 \cdot 10 = 20\)- \(2 \cdot x = 2x\)So, the expression becomes:\[20 + 2x\]

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Key Concepts

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

Understanding the Distributive Property
The distributive property is an essential tool in algebra that helps us simplify expressions. It allows us to multiply a single term by each term inside a set of parentheses.
When you see an expression like \( a(b + c) \), the distributive property tells you that this is the same as \( a imes b + a imes c \).
In our original exercise, \( 2(10 + x) \), we apply this property by multiplying \( 2 \) by \( 10 \) and \( 2 \) by \( x \).
This process helps break down more complex expressions into simpler, more manageable parts.
  • Useful for expanding expressions.
  • Makes calculations easier.
  • Helps in solving equations.
By understanding this key property, you can tackle a wide range of algebraic problems with confidence.
Breaking Down Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and operations. They are fundamental to learning algebra.
In our exercise, we start with the expression \( 2(10 + x) \), which includes a combination of a constant, \( 10 \), and a variable, \( x \).
Algebraic expressions can represent real-world situations, and simplifying them often makes them easier to work with.
  • Constants: Fixed numbers like \( 10 \) in the expression.
  • Variables: Symbols like \( x \) that can represent unknown values.
  • Operations: Addition, subtraction, multiplication, and division used to combine the numbers and variables.
Recognizing these parts helps you manipulate expressions effectively. It allows for simplification and helps solve various algebraic equations.
Mastering Multiplication in Algebra
Multiplication in algebra is an operation used to combine numbers and variables into expressions.
In the context of our original exercise, we multiply the number \( 2 \) by each part of the expression inside the parentheses: \( 10 \) and \( x \).
This results in the products \( 20 \) and \( 2x \).
  • Multiplying Constants: Simple products like \( 2 \times 10 \), resulting in \( 20 \).
  • Multiplying Variables: Operations like \( 2 \times x \), producing \( 2x \).
  • Combining Terms: Simplification involves combining these products into a single expression, \( 20 + 2x \).
Being comfortable with multiplication in algebra allows you to simplify expressions and solve more complex equations seamlessly. It's a foundational skill that, when mastered, greatly enhances your problem-solving abilities.

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Most popular questions from this chapter

Solve the equation. $$-6(-4 x-9)+4=-2(-9 x-8)$$

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