Problem 35 - Prealgebra

Question

Apply the distributive property to expression, and then simplify. \(4(5+x)\)

Step-by-Step Solution

Answer

The expression \(4(5+x)\) simplifies to \(20 + 4x\).

Step 1: Apply the Distributive Property

The distributive property states that for any numbers \(a\), \(b\), and \(c\), \(a(b + c) = ab + ac\). In the expression \(4(5+x)\), let \(a = 4\), \(b = 5\), and \(c = x\). Therefore, apply the distributive property: \(4(5) + 4(x) = 20 + 4x\).

Step 2: Simplify the Expression

The expression \(20 + 4x\) is already in its simplest form because there are no like terms to combine. \(20\) is a constant term, and \(4x\) is a linear term, so they cannot be combined further.

Key Concepts

Algebraic Expressions

An algebraic expression is a combination of numbers, variables (such as \(x\)), and operations like addition, subtraction, multiplication, and division. These expressions can represent real-world quantities. In algebra, expressions are used to construct equations and find solutions.

Take for example the expression \(4(5+x)\). This includes a numerical coefficient 4 and the variable component \(x\). The purpose of such expressions is to hold the place for any number that \(x\) may represent.

• Variables: Represent unknown quantities and can vary.
• Constants: Known numbers in an expression.
• Operations: The ways numbers and variables are combined, such as addition and multiplication.

Writing expressions correctly is the first step towards solving algebraic problems. They help us describe and model relationships between quantities.

Simplification

Simplification is the process of making an algebraic expression easier to work with by using basic arithmetic and algebraic laws, like the distributive property. This means combining like terms and reducing the expression to its most manageable form.

For the expression \(4(5+x)\), using the distributive property, we have: - Calculate \(4 \times 5\), resulting in \(20\).- Calculate \(4 \times x\), resulting in \(4x\). Combining these results gives \(20 + 4x\). Simplification involves making sure there are no like terms that can be combined further. In this case, there are no terms that share the same variable or power, so \(20 + 4x\) is already as simple as it can be.

Simplifying expressions is crucial in algebra as it makes the problems less complex and much easier to solve.

Linear Equations

Linear equations are mathematical statements that describe a straight line when graphed. They typically take the form \(ax + b = 0\) where \(a\) and \(b\) are constants and \(x\) is the variable. This format shows a relationship that changes proportionally with \(x\).

An example based on our simplified expression \(20 + 4x\) can involve setting the entire expression equal to a value, like \(20 + 4x = 0\), and solving for \(x\): - Subtract 20 from both sides to get \(4x = -20\).- Divide both sides by 4 to solve for \(x = -5\).

Linear equations are simple but fundamental in algebra. They are the building blocks for more complicated mathematical functions and model real-world scenarios where one quantity affects another in a directly proportional manner. Understanding how to manipulate and solve them is vital for progressing in math.