Problem 17 - Algebra 1: Concepts and Skills

Question

Find the product. \(-7(4)\)

Step-by-Step Solution

Answer

-28

Step 1: Identify Multiplication

The problem given \(-7(4)\) requires finding the product. Essentially this means -7 is being multiplied by 4.

Step 2: Apply negative times positive rule

According to the rules of multiplication, when a negative number and a positive number is multiplied together, the result is always negative. So multiply 7 (ignoring the minus for now) and 4.

Step 3: Find the Product

The product of 7 and 4 is 28. But remember the rule in Step 2, a negative number multiplied by a positive number must give a negative result. So, the final answer is \(-28\).

Key Concepts

Algebraic Multiplication

Algebraic multiplication is a fundamental concept in mathematics, involving the process of multiplying numbers and variables to find a product. This process is crucial as it lays the foundation for more advanced mathematical operations such as dealing with algebraic expressions and equations. Multiplication in algebra follows the same fundamental principles as multiplication with regular numbers.

When multiplying terms in algebra, each term consists of a coefficient (which can be a number) and a variable (like x or y). For example, to multiply \(3x \times 2y\), one multiplies the coefficients (3 and 2) and then combines the variables (x and y) to get \(6xy\). It's this simplicity and order that keeps algebraic multiplication consistent, allowing for easy expansion into more complex expressions.

Negative Times Positive Rule

The negative times positive rule is a key rule in multiplication that dictates the sign of the product. Understanding the sign of numbers when multiplying is just as important as the actual multiplication. The rule is simple: if you multiply a negative number by a positive number, the result will always be negative.

The reasoning behind this rule relates to the concept of direction on a number line. Positive numbers are considered to be in the 'forward' direction, while negative numbers are in the 'backward' direction. Multiplying a positive by a negative is like moving backward, hence the negative result. It’s crucial to remember this rule as it serves as the cornerstone for solving various types of algebraic problems.

Finding the Product

Finding the product means calculating the result of a multiplication operation. The product is not just about getting the answer; it’s about understanding the relationship between the factors being multiplied. For instance, in the example \( -7(4) \), the act of finding the product involves multiplying the absolute values (7 and 4) to get 28, and then applying the negative times positive rule to determine the sign of the answer.

The product in this case is \( -28 \), which reflects the application of both multiplication and the sign rule. Mastering the process of finding the product with both positive and negative numbers is essential, as these skills are used throughout algebra, physics, economics, and beyond. Students must practice with a variety of problems to build a strong foundation in this concept.