Problem 14 - Prealgebra

Question

Use the Distributive Property to write each expression as an equivalent algebraic expression. $$-3(s-7)$$

Step-by-Step Solution

Answer

The equivalent expression is \(-3s + 21\).

Step 1: Identify the Distributive Property

The Distributive Property states that for any numbers \(a\), \(b\), and \(c\), the expression \( a(b+c) \) is equal to \( ab + ac \). This property allows us to multiply a single term by each term within a parenthesis.

Step 2: Apply the Distributive Property to the Expression

The given expression is \(-3(s-7)\). According to the Distributive Property, multiply \(-3\) by both \(s\) and \(-7\). This involves two multiplications: 1. \(-3 \times s\) which gives \(-3s\).2. \(-3 \times -7\) which gives \(+21\).

Step 3: Combine the Results

Combine the results from Step 2 to find the equivalent algebraic expression. The multiplication yields an expression with two terms: \(-3s + 21\). Therefore, the equivalent expression is \(-3s + 21\).

Key Concepts

Understanding Equivalent Expressions

An equivalent expression is a rewritten version of a given expression that has the same value as the original, even though it looks different. Simplifying expressions using properties such as the distributive property, involves rewriting them in a form that might be more convenient for solving equations or understanding relationships between terms. For example, in the expression \(-3(s - 7)\), using the distributive property transformed it into \(-3s + 21\).

Both expressions may appear different but represent the same value for any given value of \(s\). Here’s why it matters:

• They allow us to solve equations more efficiently by simplifying complex expressions.
• They can be used to compare expressions directly to see if two expressions will yield the same results when solving for variable values.

Understanding equivalent expressions is fundamental because it helps in recognizing different forms of the same algebraic solution, making math feel less daunting and more consistent.

Exploring Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operation symbols. It combines these elements to depict a specific quantity or relationship. Consider the expression \(-3(s - 7)\), which includes the number -3, the variable \(s\), and the subtraction operation within the parenthesis.

Key elements of algebraic expressions include:

• Variables: Symbols, often like \(s\), that represent unknown values or can change.
• Coefficients: Numbers (e.g., -3) that multiply a variable, indicating how many times to count the variable.
• Operations: Mathematical processes such as addition, subtraction, multiplication, or division.

By manipulating these elements using properties like the distributive property, we are able to simplify complex expressions such as turning \(-3(s - 7)\) into \(-3s + 21\). This aids in making calculations more manageable when solving equations or understanding terms.

Grasping Negative Multiplication

Negative multiplication is the process of multiplying numbers with at least one negative sign. This operation is crucial in transforming expressions like \(-3(s-7)\) into \(-3s + 21\) because it demonstrates the way negatives impact the multiplication process.

Let's break down how this works:

• Multiplying a negative by a positive (e.g., \(-3 \times s\)) results in a negative product, forming \(-3s\).
• Multiplying two negatives (e.g., \(-3 \times -7\)) gives a positive result, hence \(+21\).

The rules of negative multiplication help clarify why signs change when terms are multiplied. Proper understanding of these rules ensures accurate transformation of expressions, which then allow equivalent expressions to be formed correctly. Fundamentally, recognizing these patterns in multiplication helps to maintain consistency in algebraic operations.