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Chapter 2: Problem 12

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

Short Answer

Expert verified
The equation is not linear because it contains \( y^2 \).

Step by step solution

01

Identify the Standard Form of a Linear Equation

A standard linear equation is typically written in the form \( ax + by = c \), where \( x \) and \( y \) are variables, and \( a \), \( b \), and \( c \) are constants. For the function to be linear, both variables must have an exponent of 1.
02

Analyze the Given Equation

Look at the given equation: \( 3x + 5y^2 = 15 \). Here, the equation consists of two terms involving variables: \( 3x \) and \( 5y^2 \). The term \( 5y^2 \) has the variable \( y \) raised to the power of 2.
03

Evaluate the Exponents of Variables

For a function to be linear, all the variables should be raised only to the power of 1. In \( 3x + 5y^2 = 15 \), the \( y \) variable has an exponent of 2. This means the equation is not linear.
04

Conclusion: Determine the Nature of the Equation

Since one of the variables (\( y \)) has an exponent other than 1, the equation \( 3x + 5y^2 = 15 \) is not a linear equation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Standard Form of a Linear Equation
Linear equations are fundamental in algebra and are typically expressed in the standard form: \( ax + by = c \). In this form, \( x \) and \( y \) represent variables, while \( a \), \( b \), and \( c \) are constants.
A few key things to remember about this form:
  • The equation represents a straight line when plotted on a graph.
  • The variables \( x \) and \( y \) should have exponents of 1.
  • The coefficients \( a \) and \( b \) must not both be zero.
To categorize an equation as linear, it must fit this mold. If the equation in question has variables with higher exponents, it deviates from being a linear equation. Identifying the correct form helps students understand and categorize different types of equations effectively.
Exponents in Equations
Exponents are powerful tools in math and can change the nature of an equation significantly.
When examining equations, it's crucial to look closely at the exponents assigned to the variables.
  • If all variable exponents are 1, the equation can potentially be a linear equation.
  • If a variable's exponent is higher than 1, the equation veers from linearity to a nonlinear form.
  • Nonlinear equations can include quadratic terms (e.g., \( y^2 \)), cubic terms (e.g., \( y^3 \)), and more.
In the exercise equation \( 3x + 5y^2 = 15 \), the term \( 5y^2 \) raises the variable \( y \) to the power of 2. This automatically categorizes the equation as nonlinear.
Understanding and identifying exponents are essential skills when analyzing and understanding the behavior of equations.
Analyzing Equations
The ability to analyze equations is a cornerstone skill in algebra, allowing you to determine their type and predict their behavior. Here's how you can effectively analyze an equation:
  • Check the exponents: Ensure each variable's exponent is 1 for a linear equation.
  • Look at the equation structure: Compare it to the standard linear form \( ax + by = c \).
  • Evaluate constants and coefficients: They help define the slope and y-intercept of linear equations.
With the exercise equation \( 3x + 5y^2 = 15 \), analyzing it reveals an exponent of 2 for \( y \), taking it out of the linear category.
Effectively analyzing equations allows you to quickly identify them and understand the overarching structure and behavior, a critical skill in solving mathematical problems.

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Most popular questions from this chapter

For the following exercises, which of the tables could represent a linear function? For each that could be linear, find a linear equation that models the data.

In \(2003,\) a town's population was \(1431 .\) By 2007 the population had grown to 2134 . Assume the population is changing linearly. a. How much did the population grow between the year 2003 and 2007\(?\) b. How long did it take the population to grow from 1431 people to 2134 people? c. What is the average population growth per year? d. What was the population in the year 2000 ? e. Find an equation for the population, \(P\) of the town \(t\) years after 2000 . f. Using your equation, predict the population of the town in 2014 .

Given the following set of information, find a linear equation satisfying the conditions, if possible. \(x\) intercept at \((-4,0)\) and \(y\) -intercept at \((0,-6)\)

For the following exercises, use each set of data to calculate the regression line using a calculator or other technology tool, and determine the correlation coefficient to 3 decimal places of accuracy. $$\begin{array}{|c|c|c|c|c|c|c|}\hline x & {900} & {988} & {1000} & {1010} & {1200} & {1205} \\ \hline y & {70} & {80} & {82} & {84} & {105} & {108} \\\ \hline\end{array}$$

For the following exercises, consider this scenario: The population of a city increased steadily over a ten-year span. The following ordered pairs show the population and the year over the ten-year span (population, year) for specific recorded years: $$(3,600,2000) ;(4,000,2001) ;(4,700,2003) ;(6,000,2006)$$ What is the correlation coefficient for this model to three decimal places of accuracy?
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