Problem 25 - Algebra 1

Question

DISTRIBUTIVE PROPERTY Use the distributive property to rewrite the expression without parentheses. $$ 3(x+4) $$

Step-by-Step Solution

Answer

The expression without the parentheses is \(3x + 12\).

Step 1: Identify and Apply the Distributive Property

We need to multiply both terms inside the parentheses by the number outside the parentheses. That means, multiply \(3\) by \(x\) and \(3\) by \(4\).

Step 2: Multiply the terms

After applying the distributive property, we get \(3x + 12\) as the re-written expression

Key Concepts

Algebra

Algebra is a branch of mathematics that uses symbols, typically letters, to represent numbers in equations and expressions. It allows you to create relationships and rules for working with numbers. This is essential for solving problems where numbers are unknown or variables are involved. For example, in algebra, equations such as \(3(x+4)\) can be analyzed and manipulated using properties like the distributive property.
Algebra is a fundamental part of mathematics and is often used in conjunction with other branches of math. It helps not just in academic settings, but also in everyday life to model real-world situations.

Expressions

An expression in mathematics is a combination of numbers, variables, and mathematical operations. Expressions are like sentences in math, conveying a numerical thought.
In the example \(3(x+4)\), you have an expression that includes a number \(3\), a variable \(x\), and a constant \(4\).
Unlike equations, expressions do not have an equality sign. They often need simplification or evaluation, especially when using properties like the distributive property. Understanding expressions is crucial for simplifying mathematical problems and for solving equations.

Multiplication

Multiplication is one of the basic arithmetic operations and involves adding a number, called the multiplicand, to itself a certain number of times determined by another number, called the multiplier.
For instance, multiplying \(3(x+4)\) involves distributing \(3\) across each element inside the parentheses, which means multiplying both \(3\) and \(x\) and \(3\) and \(4\).
This results in the expression \(3x + 12\).
Multiplication, especially within algebraic expressions, forms the basis for applying various properties like the distributive property, which is essential for rewriting expressions more simply and solving equations efficiently.