Problem 45 - Elementary Algebra

Question

Classify each of the equations for the following problems by degree. If the term linear, quadratic, or cubic applies, state it. $$ 3 x-7 y=9 $$

Step-by-Step Solution

Answer

Question: Classify the given equation \(3x-7y=9\) based on its highest power of the variables. Answer: Linear equation

Step 1: Identifying the highest power of each variable

In the equation \(3x-7y=9\), we can see that the highest power of both \(x\) and \(y\) is \(1\). Step 2: Classify the equation

Step 2: Classifying the equation based on the highest power

Since the highest power of the variables in this equation is \(1\), this equation is classified as a linear equation.

Key Concepts

Linear Equation

Linear equations are a foundational concept in algebra. They consist of variables raised to the first power only. The general form of a linear equation in two variables is \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants. As seen in the given example \(3x - 7y = 9\), the variables \(x\) and \(y\) are raised to the power of 1, which classifies it as linear.

Characteristics of linear equations include:
- A graph that results in a straight line when plotted on a coordinate plane
- A constant rate of change or slope
Understanding these properties is crucial as linear equations form the backbone of more complex algebraic concepts. Linear equations are used in various real-world contexts, such as calculating budgets, predicting trends, or solving problems that involve a consistent rate of change.

Degree of Equation

The degree of an equation is a critical way of classifying polynomial equations. It refers to the highest exponent of the variable in the equation. For example, the equation \(3x - 7y = 9\) has variables with the highest exponent of 1, which means it has a degree of 1.

Here's a simple breakdown of degrees:

• If only terms of the highest power are considered:
  - Degree 1: Linear (e.g., \(ax + by = c\))
  - Degree 2: Quadratic (e.g., \(ax^2 + bx + c = 0\))
  - Degree 3: Cubic (e.g., \(ax^3 + bx^2 + cx + d = 0\))

Identifying the degree is essential because it helps predict the behavior of the equation's graph and the number of solutions it may have. Remember, the degree of an equation provides insights into its nature and the type of solutions you can expect to find.

Algebraic Expression

An algebraic expression involves numbers, variables, and operations, but unlike equations, it doesn't include an equality sign. It represents a value but does not make a statement that two things are equal—unlike equations, which can be solved.

Common features of algebraic expressions include:
- They can be simplified but not "solved" like equations
- They can include various operations, such as addition, subtraction, multiplication, and division (e.g., \(3x - 7y + 9\))

Understanding algebraic expressions is key because they form the components of equations, like those seen in the problem \(3x - 7y = 9\). Mastery of working with algebraic expressions leads to improved skills in forming and solving equations, crucial for advancing in mathematical studies.