Problem 67 - Prealgebra

Question

Apply the distributive property to each of the following expressions. $$-3(5 x-1)$$

Step-by-Step Solution

Answer

-15x + 3

Step 1: Understand the Distributive Property

The distributive property states that \[ a(b + c) = ab + ac \] This means that you distribute the multiplication of \(a\) across the terms inside the parenthesis \((b + c)\).

Step 2: Identify Values

In the expression \(-3(5x - 1)\), we treat \(-3\) as \(a\), \(5x\) as \(b\), and \(-1\) as \(c\).

Step 3: Distribute the Coefficient

Using the distributive property, multiply \(-3\) by each term inside the parentheses:\[ -3(5x) = -15x \]\[ -3(-1) = 3 \]

Step 4: Combine the Results

Combine the multiplied results to form the expanded expression:\[ -15x + 3 \]

Key Concepts

Understanding Prealgebra

Prealgebra serves as the foundational step in the world of mathematics, bridging the gap between basic arithmetic and the more abstract concepts found in algebra. It is crucial because it introduces students to algebraic concepts using simpler, more accessible methods. In prealgebra, students learn about:

• Whole numbers: Counting numbers without fractions or decimals.
• Basic operations: Addition, subtraction, multiplication, and division.
• Integers: Whole numbers including negatives and zero.
• Fractions and decimals: Parts of whole numbers represented in alternate formats.
• Fundamental properties: Such as the distributive property.

Each of these elements is essential for understanding algebraic operations because they form the basis for manipulating more complex expressions.

Decoding Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and operators (like plus and minus signs) that represent a value. In algebra, understanding how to read and manipulate these expressions is essential. Let's unpack the expression \(-3(5x - 1)\). Here, \(5x - 1\) is called a binomial because it has two terms, \(5x\) and \(-1\). The expression itself is a product of \(-3\) and this binomial. Our goal is to simplify this by applying the distributive property.

Key components in mathematical expressions include:

• Terms: Individual parts of an expression, separated by plus or minus signs.
• Factors: Numbers or variables that multiply to form a term.
• Operators: Symbols that represent mathematical operations, like "+" or "\(-1\)".

Understanding each part of an expression is essential to correctly applying mathematical operations, such as the distributive property, to simplify or expand them.

The Role of Coefficients in Algebra

Coefficients are a fundamental concept in algebra, as they act as the multiplier of a variable term. They are the number in front of a variable that indicates how many times the variable is to be multiplied. For instance, in the expression \(5x\), the number \(5\) is the coefficient and \(x\) is the variable. In our expression \(-3(5x - 1)\), we distribute -3 across the terms in the parentheses. Here, \(-3\) is not just influencing the entire expression but is used to multiply each term inside, thereby distributing the coefficient effect: \(-3 \cdot 5x = -15x\) and \(-3 \cdot (-1) = 3\).Key points to remember about coefficients:

• Sign matters: A negative coefficient, as in \(-3\), changes the sign of each term it multiplies.
• Coefficients are constants: Though they multiply variables, they stay the same unless altered by operations.
• Multiplication rules: Apply multiplication across each term inside an expression when there's a coefficient outside parentheses, as dictated by the distributive property.

Understanding coefficients helps in manipulating expressions accurately and forming a strong algebraic foundation.