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Chapter 4: Problem 11

Determine whether the given equation is linear or nonlinear. $$y=3 x+2$$

Short Answer

Expert verified
Answer: The equation $$y=3x+2$$ is linear.

Step by step solution

01

Identify the variables and their degrees in the equation

In the given equation, $$y=3x+2$$, the variables are 'x' and 'y'. The degree of the variable 'x' is 1 (since it is not raised to any power), and the degree of the variable 'y' is also 1 (since it is raised to the power 0, and anything to the power 0 is considered a constant (1)).
02

Determine if the equation is linear or nonlinear

Since both variables 'x' and 'y' have a degree of 1, the equation is a linear equation. Linear equations have variables with a degree of 1, while nonlinear equations have variables with degrees other than 1. The conclusion is that the given equation $$y=3x+2$$ is a linear equation.

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Key Concepts

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

Understanding Variables
In mathematics, variables are symbols used to represent unknown values. They can change or vary, which is why they are called 'variables.' In the equation \(y = 3x + 2\), the variables are \(x\) and \(y\). These symbols stand in for numbers that can vary, allowing us to express a rule or relationship in a general way.
Variables are crucial in formulating equations and functions. In our example, \(x\) usually represents the input or independent variable, while \(y\) is the output or dependent variable. Understanding what each variable represents is vital, as it helps to interpret the equation in the context of a problem.
  • \(x\) and \(y\) are variables.
  • Variables can change or represent different numbers.
  • \(x\) is typically the independent variable, while \(y\) is dependent.
The Degree of a Term Explained
The degree of a term refers to the exponent of the variable in that term. It is an important concept when classifying equations. In the equation \(y = 3x + 2\), each term consists of a variable and its coefficient. Here鈥檚 how we determine their degrees:
- The term \(3x\) has 'x' raised to the power of 1. Therefore, its degree is 1.- The constant term \(2\) does not have a variable associated with it, so its degree is considered 0.
The degree gives us essential information about the highest power of the variable in the equation. For a linear equation, like the one we're examining, the variables need to have a degree of 1.
  • Terms like \(3x\) have a degree of 1.
  • Constant terms, like \(2\), have a degree of 0.
  • The highest degree in any term determines the degree of the equation.
Different Types of Equations
Equations can be broadly classified into linear and nonlinear equations based on the degree of their terms. A linear equation is any equation where each term has a degree of 1. This makes the graph of a linear equation a straight line.
In contrast, a nonlinear equation would have at least one term with a degree other than 1. This would result in a graph that is not a straight line. Nonlinear equations can include quadratic, cubic, or even more complex equations where the variable is squared, cubed, etc.
In the example \(y = 3x + 2\), since both variables, \(x\) and \(y\), are to the first power, the equation is classified as linear. This is because:
  • All variable terms have a degree of 1.
  • The graph of this equation will be a straight line.
  • Nonlinear equations have degrees of 2 (quadratic) or higher (cubic, etc.)

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Most popular questions from this chapter

For Exercises \(25-30,\) explain the mistake in the translation. Then write the correct translation. Negative three times the sum of \(y\) and five is equal to twice \(y\) minus the difference of \(y\) and one. Translation: \(-3(y+5)=2 y-(1-y)\)

Solve for the unknown amount. The Star Spangled Banner flown over Fort McHenry during the Battle of Baltimore in 1814 for which the song that became the U.S. national anthem was written has an area of 1260 square feet. If the width of the flag is 30 feet, what is the length? (Use \(A=l w .\) )

Check to see if the given number is a solution for the given equation. $$3 m+10=-2 m ; \text { check } m=5$$

Check to see if the given number is a solution for the given equation. $$5 d-1=19 ; \text { check } d=4$$

Solve and check. $$-6=y+19$$
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