Problem 11 - College Algebra with Corequisite Support

Question

For the following exercises, determine whether the equation of the curve can be written as a linear function. $$ 3 x+5 y^{2}=15 $$

Step-by-Step Solution

Answer

The equation cannot be written as a linear function due to the \( y^2 \) term.

Step 1: Reorganize the Original Equation

Start by expressing the given equation in a form where you can easily determine if it can be written as a linear equation. The given equation is \( 3x + 5y^2 = 15 \). A linear function in two variables typically has the form \( y = mx + b \), where \( y \) is raised to the first power.

Step 2: Consider the Terms Separately

Examine the terms in the given equation: \( 3x \) and \( 5y^2 \). Notice that the term \( 5y^2 \) includes \( y^2 \), which is a quadratic term. Linear functions only include first-degree (exponent of 1) terms. The presence of \( y^2 \) suggests this is not a linear equation.

Step 3: Conclusion about the Nature of the Equation

Since the term \( 5y^2 \) is quadratic, the given equation cannot be transformed into a standard linear equation. Linear functions do not include squared terms, which makes this equation non-linear.

Key Concepts

Quadratic Equation

A quadratic equation is a mathematical expression of the form \( ax^2 + bx + c = 0 \). The key feature that classifies an equation as quadratic is the presence of the term \( x^2 \), which indicates the equation has a degree of two. Quadratic equations are crucial in various fields like physics and engineering.

When dealing with quadratic equations, it’s important to look for the squared term. In our given equation \( 3x + 5y^2 = 15 \), the \( 5y^2 \) term highlights that we have a quadratic component because of the \( y^2 \). Unlike linear equations, quadratic equations can take the shape of a parabola when graphed.

Understanding how to distinguish quadratic terms helps when determining the nature of equations and whether they represent linear functions or more complex relationships like quadratic functions.

Non-linear Function

A non-linear function is defined as any function that doesn’t create a straight line when graphed. This includes equations where the variable is raised to a power other than one, or where different types of terms are multiplied together.

In the context of our exercise, the equation \( 3x + 5y^2 = 15 \) contains the term \( 5y^2 \), which makes the function non-linear because \( y \) is raised to the second power. This violates the definition of a linear function, which only permits variables to the first degree. Nonlinear functions, such as quadratic functions, often exhibit curves, peaks, or valleys unlike the straight trajectory of linear functions.

Learning to quickly identify non-linear characteristics in equations can assist in analyzing more complex mathematical relationships and understanding the limitations of linear models.

Algebraic Expressions

Algebraic expressions consist of numbers, symbols, and operators (like \( +, -, \times, \div \)) that define a mathematical statement or equation. They are the building blocks of algebra and include variables, which are symbols representing unknown values.

In our equation \( 3x + 5y^2 = 15 \), each part, such as \( 3x \) and \( 5y^2 \), is an algebraic expression. The linear part \( 3x \) consists of a single variable raised to the first power, typical of linear functions. In contrast, \( 5y^2 \) is a non-linear algebraic expression due to the squared variable, which adds complexity to solving or graphing the equation.

Grasping the differences between various algebraic expressions helps students in solving equations and understanding the foundations of algebra, making it easier to identify patterns and solutions in more intricate mathematics problems.