What Is The Slope Of The Line Represented By The Equation Y = − 2 3 − 5 X Y = -\frac{2}{3} - 5x Y = − 3 2 − 5 X ?A. -5 B. -\frac{2}{3} C. \frac{2}{3} D. 5
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. The slope is often denoted by the letter 'm' and is a key component of the slope-intercept form of a linear equation, which is given by:
y = mx + b
where 'm' is the slope, 'x' is the independent variable, and 'b' is the y-intercept.
Determining the Slope of a Line
To determine the slope of a line represented by the equation y = -\frac{2}{3} - 5x, we need to identify the coefficient of the x-term. In this case, the coefficient of the x-term is -5. The slope of the line is therefore -5.
Why is the Slope -5?
The slope of the line is -5 because the coefficient of the x-term is -5. This means that for every unit increase in the x-coordinate, the y-coordinate decreases by 5 units. In other words, the line is decreasing at a rate of 5 units per unit increase in the x-coordinate.
Comparing the Options
Now that we have determined the slope of the line to be -5, let's compare it with the given options:
A. -5 B. -\frac{2}{3} C. \frac{2}{3} D. 5
Based on our analysis, the correct answer is:
A. -5
Conclusion
In conclusion, the slope of the line represented by the equation y = -\frac{2}{3} - 5x is -5. This is because the coefficient of the x-term is -5, which represents the rate of change of the line. Understanding the concept of slope is essential in mathematics, and it has numerous applications in various fields, including physics, engineering, and economics.
Additional Tips and Tricks
- When determining the slope of a line, always identify the coefficient of the x-term.
- The slope of a line can be positive, negative, or zero.
- A positive slope indicates that the line is increasing, while a negative slope indicates that the line is decreasing.
- A slope of zero indicates that the line is horizontal.
Real-World Applications of Slope
Slope has numerous real-world applications, including:
- Physics: The slope of a line can be used to represent the acceleration of an object.
- Engineering: The slope of a line can be used to design and build structures, such as bridges and buildings.
- Economics: The slope of a line can be used to represent the relationship between two variables, such as supply and demand.
Common Mistakes to Avoid
- Mistaking the slope for the y-intercept: The slope and y-intercept are two distinct components of a linear equation.
- Not identifying the coefficient of the x-term: The coefficient of the x-term is the key to determining the slope of a line.
- Not considering the sign of the slope: The sign of the slope is crucial in determining the direction of the line.
Conclusion
Q: What is the slope of a line?
A: 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.
Q: How do I determine the slope of a line?
A: To determine the slope of a line, you need to identify the coefficient of the x-term in the equation. The coefficient of the x-term is the key to determining the slope of a line.
Q: What is the difference between the slope and the y-intercept?
A: The slope and y-intercept are two distinct components of a linear equation. The slope represents the rate of change of the line, while the y-intercept represents the point where the line intersects the y-axis.
Q: Can the slope of a line be positive, negative, or zero?
A: Yes, the slope of a line can be positive, negative, or zero. A positive slope indicates that the line is increasing, while a negative slope indicates that the line is decreasing. A slope of zero indicates that the line is horizontal.
Q: How do I use the slope to determine the direction of a line?
A: To determine the direction of a line, you need to consider the sign of the slope. If the slope is positive, the line is increasing. If the slope is negative, the line is decreasing. If the slope is zero, the line is horizontal.
Q: Can the slope of a line be a fraction?
A: Yes, the slope of a line can be a fraction. For example, the slope of the line y = \frac{2}{3}x is \frac{2}{3}.
Q: How do I use the slope to solve real-world problems?
A: The slope can be used to solve a wide range of real-world problems, including:
- Physics: The slope of a line can be used to represent the acceleration of an object.
- Engineering: The slope of a line can be used to design and build structures, such as bridges and buildings.
- Economics: The slope of a line can be used to represent the relationship between two variables, such as supply and demand.
Q: What are some common mistakes to avoid when working with slope?
A: Some common mistakes to avoid when working with slope include:
- Mistaking the slope for the y-intercept: The slope and y-intercept are two distinct components of a linear equation.
- Not identifying the coefficient of the x-term: The coefficient of the x-term is the key to determining the slope of a line.
- Not considering the sign of the slope: The sign of the slope is crucial in determining the direction of the line.
Q: How do I graph a line using its slope and y-intercept?
A: To graph a line using its slope and y-intercept, you need to follow these steps:
- Plot the y-intercept: The y-intercept is the point where the line intersects the y-axis.
- Use the slope to determine the direction of the line: The slope can be used to determine the direction of the line.
- Plot additional points: Additional points can be plotted using the slope and y-intercept.
- Draw the line: The line can be drawn by connecting the plotted points.
Q: Can the slope of a line be used to determine the equation of a line?
A: Yes, the slope of a line can be used to determine the equation of a line. The equation of a line can be written in the form y = mx + b, where m is the slope and b is the y-intercept.
Conclusion
In conclusion, the slope of a line is a fundamental concept in mathematics that represents the rate of change of a linear equation. By understanding the concept of slope, you can determine the direction of a line, solve real-world problems, and graph lines using their slope and y-intercept.