Problem 40 - Elementary Algebra

Question

Classify each of the equations for the following problems by degree. If the term linear, quadratic, or cubic applies, state it. $$ 9 z=12 x-18 $$

Step-by-Step Solution

Answer

Question: Classify the given equation by degree and state whether it is linear, quadratic, or cubic. The equation is: $$ 9z = 12x - 18 $$ Answer: The given equation is of degree 1, so it is a linear equation.

Step 1: Identify the highest degree term

Observe the given equation and identify the terms with their respective degrees. The given equation is: $$ 9z = 12x - 18 $$ The equation has three terms: \(9z\), \(12x\), and \(-18\). The degrees of the terms are as follows: - Degree of \(9z\): 1 (since \(z^1\)) - Degree of \(12x\): 1 (since \(x^1\)) - Degree of \(-18\): 0 (since it is a constant)

Step 2: Classify the equation by degree

The highest degree among the terms is 1. Therefore, the given equation is of degree 1.

Step 3: State the term if linear, quadratic, or cubic

Since the equation is of degree 1, it is a linear equation.

Key Concepts

Understanding Linear Equations

A linear equation is one of the most fundamental concepts in algebra. It represents a straight line when graphed on a coordinate plane and is characterized by its simplicity and solvability. For an equation to be classified as linear, it must follow the standard form:

\( Ax + By = C \), where \( A \), \( B \), and \( C \) are constants, and \( x \) and \( y \) are variables.
In the particular example \( 9z = 12x - 18 \), despite it having variables \( x \) and \( z \), it retains the characteristics of a linear equation as none of the terms are raised to the power higher than one.

Equation Degrees Demystified

Identifying the Degree

The degree of an equation is identified by the highest power of the variable in any term of the algebraic expression. For instance, a quadratic equation has a degree of 2, represented by the standard form \( ax^2+bx+c=0 \), where \( a \), \( b \), and \( c \) are constants. Equations of degree 1 are linear, those of degree 2 are quadratic, and those of degree 3 are called cubic.
In our exercise, the highest power of any variable is 1, indicative of a linear equation. This notion of degree is vital for students to grasp as it dictates the methods used for solving the equation as well as the shape of its graph.

Algebraic Expressions Overview

Building Blocks of Algebra

Algebraic expressions are the building blocks of algebra and are composed of numbers, variables (like \( x \) or \( z \)), and operation signs (addition, subtraction, multiplication, etc.). The complexity of an algebraic expression can vary greatly, from a simple constant like \( 5 \) to a more complex equation like \( 9z = 12x - 18 \).
It's crucial for learners to understand how to work with these expressions, as they form the basis for topics ranging from solving simple equations to more advanced mathematics like calculus. The expression provided in the exercise is a linear algebraic expression, as it forms a linear equation when two expressions are set equal to each other.