Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 4: Problem 7
For the following exercises, determine whether the equation of the curve can be written as a linear function. $$ y=3 x-5 $$
Short Answer
Expert verified
Yes, the equation \( y = 3x - 5 \) is a linear function.
Step by step solution
01
Identify the Equation
The given equation is \( y = 3x - 5 \). This equation needs to be checked if it represents a linear function.
02
Recall the Format of a Linear Equation
A linear equation in two variables is typically in the format \( y = mx + b \), where \( m \) represents the slope and \( b \) represents the y-intercept.
03
Compare Equation to Linear Form
Comparing \( y = 3x - 5 \) to \( y = mx + b \), we see that this fits the linear form with \( m = 3 \) and \( b = -5 \).
04
Analyze the Characteristics of a Linear Function
A linear function forms a straight line when graphed, has a constant rate of change, and its variable (\( x \)) is raised only to the power of 1.
05
Confirm the Equation is Linear
Since the given equation \( y = 3x - 5 \) aligns with the form \( y = mx + b \) and possesses characteristics of a linear function, it can indeed be written as a linear function.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding the Slope in Linear Equations
The slope is a crucial concept in linear equations. It is often symbolized by the letter \( m \) in the equation format \( y = mx + b \). The slope represents the steepness or incline of a line on a graph. More technically, it describes how much \( y \) (the vertical axis) changes for a given change in \( x \) (the horizontal axis).
Here are some key aspects of the slope:
Here are some key aspects of the slope:
- Positive Slope: If the slope is positive, the line rises as it moves from left to right.
- Negative Slope: If the slope is negative, the line falls as it moves from left to right.
- Zero Slope: If the slope is zero, the line is perfectly horizontal, indicating no change in \( y \) when \( x \) changes.
- Undefined Slope: If a line is vertical, the slope is undefined since \( x \) doesn't change.
The Role of the y-intercept
The y-intercept is another essential component of a linear equation, denoted by \( b \) in the equation \( y = mx + b \). It signifies the point where the line crosses the y-axis. In simpler terms, when \( x = 0 \), \( y \) is the y-intercept.
Here's what you need to know about the y-intercept:
In our example, \( y = 3x - 5 \), the y-intercept is -5. So, the line crosses the y-axis at the point \( (0, -5) \). Understanding the y-intercept helps in quickly sketching the line on a graph.
Here's what you need to know about the y-intercept:
- Initial Value: It represents the initial value or starting point of the line at the y-axis.
- Graphical Representation: The y-intercept is the point \( (0, b) \) on a graph.
In our example, \( y = 3x - 5 \), the y-intercept is -5. So, the line crosses the y-axis at the point \( (0, -5) \). Understanding the y-intercept helps in quickly sketching the line on a graph.
Recognizing a Linear Function
A linear function like \( y = mx + b \) is characterized by a straight line graph. It is a simple but powerful mathematical representation widely used to describe relationships where the rate of change is constant. Here are the defining traits of a linear function:
- Straight Line: The graph forms a straight line.
- Consistent Change: Variables change at a constant rate.
- First Power: The variable \( x \) is only raised to the power of 1.
Exploring the Rate of Change
The rate of change is a fundamental concept linked to the slope in linear functions. It quantifies how much one quantity changes with respect to another. In the context of a linear equation, the rate of change is constant, meaning it remains the same regardless of the points chosen on the line.
Consider these important elements of the rate of change:
In our equation \( y = 3x - 5 \), the rate of change is 3, indicating a uniform increase of \( y \) for every unit change in \( x \). This steady rate of change is what makes linear functions both simple and powerful in modeling direct relationships.
Consider these important elements of the rate of change:
- Constant Nature: In linear equations, this constancy is expressed by the slope \( m \).
- Comparison: It compares the change in \( y \) (dependent variable) to the change in \( x \) (independent variable).
- Practical Applications: Understanding rates of change can help in predicting trends and making forecasts in various fields, such as economics or physics.
In our equation \( y = 3x - 5 \), the rate of change is 3, indicating a uniform increase of \( y \) for every unit change in \( x \). This steady rate of change is what makes linear functions both simple and powerful in modeling direct relationships.
