What Is The Slope Of The Line Represented By The Equation Y = − 1 2 X + 1 4 Y=-\frac{1}{2} X+\frac{1}{4} Y = − 2 1 X + 4 1 ?A. − 1 2 -\frac{1}{2} − 2 1 B. − 1 4 -\frac{1}{4} − 4 1 C. 1 4 \frac{1}{4} 4 1 D. 1 2 \frac{1}{2} 2 1
Introduction
In mathematics, the slope of a line is a fundamental concept that represents the rate of change of a linear equation. It is a crucial aspect of understanding the behavior of lines and their relationships with other geometric shapes. In this article, we will delve into the concept of slope and explore how to determine the slope of a line represented by a given equation.
What is Slope?
The slope of a line is a measure of how steep it is. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. In mathematical terms, the slope (m) of a line is given by the formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are two points on the line.
The Equation of a Line
The equation of a line can be written in the form:
y = mx + b
where m is the slope of the line, x is the independent variable, y is the dependent variable, and b is the y-intercept.
Slope in the Equation
Now, let's analyze the given equation:
y = -\frac{1}{2} x + \frac{1}{4}
In this equation, the slope (m) is represented by the coefficient of x, which is -\frac{1}{2}. The y-intercept (b) is \frac{1}{4}.
Determining the Slope
To determine the slope of the line represented by the equation, we need to identify the coefficient of x. In this case, the coefficient of x is -\frac{1}{2}, which represents the slope of the line.
Conclusion
In conclusion, the slope of the line represented by the equation is -\frac{1}{2}. This means that for every unit increase in x, the value of y decreases by \frac{1}{2} unit.
Answer
The correct answer is:
A.
Additional Examples
To further illustrate the concept of slope, let's consider a few more examples:
- The equation of a line with a slope of 2 and a y-intercept of 3 is: y = 2x + 3
- The equation of a line with a slope of -3 and a y-intercept of 2 is: y = -3x + 2
- The equation of a line with a slope of 0 and a y-intercept of 5 is: y = 0x + 5
Real-World Applications
Understanding the concept of slope has numerous real-world applications, including:
- Physics: The slope of a line can represent the rate of change of velocity or acceleration.
- Engineering: The slope of a line can be used to design and optimize systems, such as bridges and buildings.
- Economics: The slope of a line can represent the rate of change of economic variables, such as GDP and inflation.
Conclusion
Q1: What is the slope of a line?
A1: The slope of a line is a measure of how steep it is. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line.
Q2: How do I calculate the slope of a line?
A2: To calculate the slope of a line, you can use the formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are two points on the line.
Q3: What is the difference between slope and y-intercept?
A3: The slope of a line represents the rate of change of the line, while the y-intercept represents the point where the line intersects the y-axis.
Q4: How do I determine the slope of a line from its equation?
A4: To determine the slope of a line from its equation, you need to identify the coefficient of x. The coefficient of x represents the slope of the line.
Q5: What is the slope of a horizontal line?
A5: The slope of a horizontal line is 0, since there is no vertical change (rise) between any two points on the line.
Q6: What is the slope of a vertical line?
A6: The slope of a vertical line is undefined, since there is no horizontal change (run) between any two points on the line.
Q7: Can the slope of a line be negative?
A7: Yes, the slope of a line can be negative. A negative slope indicates that the line slopes downward from left to right.
Q8: Can the slope of a line be zero?
A8: Yes, the slope of a line can be zero. A slope of zero indicates that the line is horizontal.
Q9: Can the slope of a line be undefined?
A9: Yes, the slope of a line can be undefined. An undefined slope indicates that the line is vertical.
Q10: Why is understanding slope important?
A10: Understanding slope is important because it allows us to analyze and interpret the behavior of lines and their relationships with other geometric shapes. It has numerous real-world applications in physics, engineering, economics, and more.
Additional Resources
For further learning and practice, we recommend the following resources:
- Khan Academy: Slope and Linear Equations
- Mathway: Slope Calculator
- Wolfram Alpha: Slope and Linear Equations
Conclusion
In conclusion, understanding slope is a fundamental concept in mathematics that has numerous real-world applications. By answering these frequently asked questions, we hope to have provided a comprehensive overview of the concept of slope and its importance in mathematics and beyond.