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Chapter 4: Problem 11

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 due to the \(y^2\) term.

Step by step solution

01

Identify the General Form of a Linear Equation

A linear equation in two variables can be written in the form \(Ax + By = C\), where \(A\), \(B\), and \(C\) are constants, and both \(x\) and \(y\) are to the first power only.
02

Examine the Given Equation

The given equation is \(3x + 5y^2 = 15\). Observe the terms in the equation to determine if they all satisfy the conditions of a linear equation.
03

Check the Power of Each Variable

In the equation \(3x + 5y^2 = 15\), the variable \(x\) is raised to the first power. However, the variable \(y\) is raised to the second power (\(y^2\)), which is not allowed for a linear equation.
04

Determine the Nature of the Equation

Since the term \(5y^2\) has \(y\) raised to the power of 2, the equation is not linear. Linear equations require that all variables are only raised to the first power.

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Key Concepts

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

Linear Function
A linear function is a fundamental concept in algebra. It's an equation that produces a straight line when plotted on a graph. The most basic form of a linear function is expressed as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Linear functions have specific characteristics:
  • They involve two variables, typically \(x\) and \(y\).
  • Both variables are to the first power.
  • They do not include variables with exponents greater than one, fractions involving variables, or products of variables.
Understanding linear functions is crucial because they model many real-world situations where there is a constant rate of change.
Non-Linear Equation
A non-linear equation is one that does not form a straight line when graphed. Instead, its graph could be a curve or some other form. Non-linear equations have a few distinguishing features:
  • They may include terms with variables raised to a power other than one.
  • The variables may be multiplied together or included within more complex functions like square roots or exponential functions.
Non-linear equations include quadratic equations, cubic equations, and many more complex forms. In the equation \(3x + 5y^2 = 15\), the presence of \(y^2\) makes it non-linear, as the power of the variable \(y\) exceeds one.
Two Variables
Equations with two variables, often termed as bivariate equations, are central to the study of algebra. These equations can describe a relationship between two changing quantities. In a typical linear equation of the form \(Ax + By = C\), both \(A\) and \(B\) are constants, and \(x\) and \(y\) are variables.
  • The value of one variable depends on the other, lending the equation its dynamic property.
  • Solving involves finding value pairs \((x, y)\) that satisfy the equation.
Through graphing, pairs that solve the equation form a pattern, such as a line for linear equations, helping to visualize the relationship between variables.
Power of a Variable
The power of a variable in an equation indicates the degree to which the variable is raised. It's a key factor in determining the form and complexity of an equation.
  • If a variable is raised to the first power, it's linear in nature, contributing to a linear equation.
  • When the power exceeds one, such as \(y^2\) in \(3x + 5y^2 = 15\), the equation becomes non-linear.
The power defines how the variable influences the shape and solution of the equation. For example, a variable squared creates a parabolic curve, while a cubed term might result in more complex curvature. Recognizing the power of variables helps to classify and solve equations effectively.

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Most popular questions from this chapter

For the following exercises, use the functions \(f(x)=-0.1 x+200\) and \(g(x)=20 x+0.1\). Where is \(f(x)\) greater than \(g(x) ?\) Where is \(g(x)\) greater than \(f(x)\) ?

For the following exercises, sketch a line with the given features. An \(x\) -intercept of (-4,0) and \(y\) -intercept of (0,-2)

Graph the linear function \(f\) on a domain of [-10,10] for the function whose slope is \(\frac{1}{8}\) and \(y\) -intercept is \(\frac{31}{16}\). Label the points for the input values of -10 and 10 .

For the following exercises, consider this scenario: The number of people afflicted with the common cold in the winter months steadily decreased by 205 each year from 2005 until 2010 . In \(2005,12,005\) , \(12,025\) people were afflicted. If the function \(C\) is graphed, find and interpret the slope of the function.

For the following exercises, find the \(x\) - and \(y\) -intercepts of each equation. $$ k(x)=-5 x+1 $$
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