Problem 44 - Elementary Algebra

Question

Classify each of the equations for the following problems by degree. If the term linear, quadratic, or cubic applies, state it. $$ 2 y+5 x-3+4 x y=5 x y+2 y $$

Step-by-Step Solution

Answer

Answer: The equation is classified as linear.

Step 1: Simplify the equation

Subtract 5xy and 2y from both sides of the equation: $$ 2y + 5x - 3 + 4xy - 5xy - 2y = 5xy + 2y - 5xy - 2y $$ $$ 5x - 3 - xy = 0 $$

Step 2: Identify the highest power of variables

In the simplified equation, we see that the highest power of any variable is 1. The term with the highest power is -xy.

Step 3: Classify the equation based on the highest power

Since the highest power of any variable in the simplified equation is 1, we classify the equation as linear.

Key Concepts

Linear Equations

Linear equations are fundamental in algebra and are characterized by the highest power of the variable being one. These equations form straight lines when graphed on a coordinate plane. The general form of a linear equation in two variables, x and y, is given by \( ax + by = c \) where \( a \) and \( b \) are coefficients and \( c \) is a constant.

In our example, after simplification, we arrived at \( 5x - 3 - xy = 0 \) which, despite containing two variables, is classified as a linear equation because the terms involve only the first power of the variables or are constants. Unlike quadratic or cubic equations, linear equations will not have curved graphs, and their solutions represent points where the line intersects the axes.

Equation Simplification

Simplification of equations is a crucial step in solving and understanding them. This process involves combining like terms, eliminating redundancies, and simplifying expressions to make the equation more manageable and clearer. It often involves performing arithmetic operations and applying properties of equality to both sides of the equation.

In the provided exercise, the equation \( 2y + 5x - 3 + 4xy - 5xy - 2y = 5xy + 2y - 5xy - 2y \) simplifies to \( 5x - 3 - xy = 0 \) because we subtract terms like \( 5xy \) and \( 2y \) from both sides to eliminate them, leading to a more concise form. This process is essential for clearly identifying the degree of the equation and allows for easier classification and solution.

Degree of Polynomial

The degree of a polynomial equation is the highest power to which the variable is raised in any term of the polynomial. It determines the equation's basic shape when graphed and the number of possible solutions or roots the equation can have.

Polynomials are classified into different types based on their degree. For example, a first-degree polynomial is linear, a second-degree is quadratic, and a third-degree is cubic. Our example resulted in a simplified polynomial \( 5x - 3 - xy = 0 \) with the highest power of one, making it a first-degree polynomial, or more specifically, a linear polynomial.