Problem 10 - Prealgebra

Question

Simplify expression. \(x+3(x+4 y)\)

Step-by-Step Solution

Answer

The simplified expression is \(4x + 12y\).

Step 1: Identify the expression to be simplified

The given expression is \(x + 3(x + 4y)\). Our goal is to simplify this expression by removing any parentheses and combining like terms.

Step 2: Distribute the constant through the parentheses

Begin by distributing the 3 across the terms inside the parentheses: \(3(x + 4y)\). Use the distributive property: \(3x + 3(4y) = 3x + 12y\).

Step 3: Rewrite the expression

Substitute the distributed terms back into the original expression: \(x + 3x + 12y\).

Step 4: Combine like terms

Combine the terms that contain \(x\): \(x + 3x\). This results in \(4x\).

Step 5: Write the simplified expression

The simplified form of the expression is \(4x + 12y\).

Key Concepts

Distributive Property

The distributive property is a key concept in algebra that helps us simplify expressions involving parentheses. It allows you to "distribute" a multiplier over terms within a set of parentheses. For instance, in the expression \(3(x + 4y)\), the number 3 needs to be multiplied by each term inside the parentheses, namely \(x\) and \(4y\).
In step-by-step terms:

• Multiply the 3 with the \(x\) yielding \(3x\).
• Next, multiply the 3 with the \(4y\) yielding \(12y\).

After applying the distributive property, the expression \(3(x + 4y)\) becomes \(3x + 12y\).
This process eliminates the parentheses, making it easier to move forward with combining terms. The distributive property is often used to simplify complex algebraic equations and is a fundamental component of many algebraic operations.

Like Terms

Understanding like terms is essential when simplifying algebraic expressions. Like terms are terms that have identical variable parts raised to the same power. In other words, their variable components are the same.
For example:

• \(x\) and \(3x\) are like terms because they share the same variable \(x\).
• \(12y\) doesn't have any like terms with \(x\) or \(3x\) because it is associated with a different variable, \(y\).

To simplify an expression, you must identify and combine like terms. This means adding or subtracting the coefficients (the numbers in front of the variables) of like terms.
For instance, when simplifying \(x + 3x + 12y\), you would add \(x\) and \(3x\) to get \(4x\), while \(12y\) remains unchanged because there are no other like terms to combine it with.

Expression Simplification

Expression simplification is the process of reducing an algebraic expression to its simplest form. Simplifying an expression often involves a few key steps which make the expression easier to read and work with.
Here’s how simplification works with the expression \(x + 3(x + 4y)\):

• Start by applying the distributive property to eliminate parentheses, which gives you \(x + 3x + 12y\).
• Next, identify and combine any like terms. In this case, that's combining \(x\) and \(3x\) to get \(4x\).
• The resultant expression \(4x + 12y\) is the simplified version of the original.

Simplifying expressions allows for easier manipulation of the equations in solving or analyzing problems. This skill is foundational in algebra and crucial for advancing in mathematics.