Problem 17 - Elementary Algebra

Question

For the following problems, a term will be given followed by a group of its factors. List the coefficient of the given group of factors. $$ 8 z, z $$

Step-by-Step Solution

Answer

Answer: The coefficient is 8.

Step 1: Understand the given term and its factors

We are given the following term: $$ 8z $$ And the group of factors: $$ z $$

Step 2: Identify the coefficient in the given term

A coefficient is a numerical value that is multiplied by a variable in an algebraic expression. In this case, the coefficient is the number in front of the variable \(z\). Here, the coefficient of the \(z\) is \(8\). So, the term looks like: $$ 8z = 8 \cdot z $$

Step 3: Find the coefficient of the given group of factors

We already have the coefficient and the group of factors, we just need to express it as desired. The expression is: $$ 8z $$ The group of factors is: $$ z $$ And the coefficient of the given group of factors is \(8\). The answer is \(8\).

Key Concepts

Algebraic Expressions

Algebraic expressions are the building blocks of algebra and combine numbers, variables, and operation symbols to form meaningful mathematical statements. They can be as simple as a single term, like 4x, or as complex as a multi-term expression, such as 4x^2 + 3xy - 5. In algebraic expressions, the order of operations matters, so it's important to evaluate them correctly using the PEMDAS rule, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

Understanding how to deconstruct an algebraic expression into its components helps students solve equations and understand functions. For instance, in the term 8z, we have a simple algebraic expression consisting of a coefficient, a variable, and an implied multiplication between them. The number 8 is the coefficient, which quantifies how many times the variable z is included in the expression.

Factors in Algebra

Factors in algebra are the elements that, when multiplied together, produce an algebraic expression. When we refer to 'factoring' an expression, we are looking for a way to break it down into these multiplicative components. For example, if we consider the algebraic term 8z, we say that 8 and z are factors because when multiplied, they produce the original term. Factoring is a key concept in algebra as it allows us to simplify expressions, solve equations, and understand the properties of numbers and terms.

In the context of our exercise, we focused on identifying the coefficient 8 when given the factor z. Coefficients are often the numeric factors of terms that provide a multiplier for the variables they accompany.

Variables in Algebra

In algebra, variables are symbols, usually letters, that represent unknown values. They can take on any value, and we use them to generalize math problems and create formulas. Variables like x, y, and z are common placeholders for these unknown quantities. In an expression such as 8z, z is the variable, which suggests that there could be 8 of some quantity, but we don't specify what that quantity is.

The beauty of using variables lies in the ability to manipulate equations and expressions to solve for them. When a problem asks for the coefficient of a given group of factors, it essentially asks us to identify the numeric part of the term associated with the variable or variables present. In our exercise, identifying that the coefficient of z in 8z is 8 helps us understand how to construct and deconstruct expressions with variables to find solutions in algebra.