Problem 10 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

No, the equation is not linear because it contains \( x^2 \).

Step 1: Identify Linear Function Characteristics

A linear function can be written in the form \( y = mx + b \), where \( m \) and \( b \) are constants, and \( x \) and \( y \) are variables. The highest degree of any variable in a linear equation is 1.

Step 2: Examine the given equation

The given equation is \( 3x^2 + 5y = 15 \). It contains a term \( 3x^2 \), which has a degree of 2.

Step 3: Determine if the equation is linear

Since the term \( 3x^2 \) has a degree of 2, the equation does not fit the form of \( y = mx + b \) because a linear equation cannot have terms with exponents other than 1 or constants. Thus, this is not a linear function.

Key Concepts

Equation Characteristics

Understanding the characteristics of an equation is crucial to determine its type. A linear equation is typically expressed in the form of \( y = mx + b \), where both \( m \) (the slope) and \( b \) (the y-intercept) are constants. The variables \( x \) and \( y \) are raised only to the first power. This means:

• The equation graph forms a straight line;
• The highest degree of variables is 1;
• There are no products or quotients of variables involved;
• All coefficients are constant, meaning they do not change.

Recognizing these attributes helps in identifying whether an equation is linear or not. Consequently, a true linear equation will form a straight line graph when plotted.

Non-linear Equations

Non-linear equations, unlike linear equations, include terms where the variables have exponents other than 1. These could be quadratic (like \( x^2 \)), cubic (like \( x^3 \)), or even involve more complex expressions such as \( \sqrt{x} \) or \( \frac{1}{x} \). Such equations represent:

• Curved graphs rather than straight lines;
• Degrees of terms higher than one;
• Potential for intersections, turning points, and varying slopes.

In the example \( 3x^2 + 5y = 15 \), the term \( 3x^2 \) is a clear indicator of a non-linear equation due to its degree of 2. Identifying higher degrees in variables points towards a non-linear nature.

Linear Equation Form

The most simple form of a linear equation is the slope-intercept form: \( y = mx + b \). Here, every term follows specific rules:

• \( m \), the slope, shows the angle of the line;
• \( b \) is the y-intercept where the line crosses the y-axis;
• No variable is multiplied by another, nor are there any powers greater than one on those variables.

This form describes every linear equation's core structure, laying a foundation for understanding more complex mathematical contexts. If an equation does not adhere to this structure, such as including terms like \( x^2 \), the equation cannot be considered as a linear function. Therefore, knowing this form is key to identifying linear relationships.