What Is The Slope Of The Line Represented By The Equation $y=\frac{2}{3}-5x$?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 various mathematical concepts, including algebra, geometry, and calculus. 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 other words, it represents the rate at which the line rises or falls as you move from left to right.
The Equation of a Line
The equation of a line is typically written in the form y = mx + b, where:
- m is the slope of the line
- x is the independent variable (the input or the horizontal axis)
- y is the dependent variable (the output or the vertical axis)
- b is the y-intercept (the point where the line intersects the y-axis)
Determining the Slope of a Line
To determine the slope of a line represented by the equation y = mx + b, we need to identify the value of m. In the given equation y = 2/3 - 5x, we can see that the slope is represented by the coefficient of x, which is -5.
Slope in the Given Equation
Let's take a closer look at the given equation y = 2/3 - 5x.
- The coefficient of x is -5, which represents the slope of the line.
- The constant term 2/3 is the y-intercept, which is the point where the line intersects the y-axis.
Calculating the Slope
To calculate the slope of the line, we can use the formula:
m = (y2 - y1) / (x2 - x1)
However, in this case, we are given the equation of the line in the form y = mx + b, so we can directly identify the slope as the coefficient of x.
Conclusion
In conclusion, the slope of the line represented by the equation y = 2/3 - 5x is -5. This means that for every unit increase in x, the value of y decreases by 5 units.
Answer
The correct answer is A. -5.
Additional Tips and Resources
- To determine the slope of a line, you can use the formula m = (y2 - y1) / (x2 - x1) or identify the coefficient of x in the equation of the line.
- The slope of a line can be positive, negative, or zero, depending on the direction and steepness of the line.
- For more information on slope and other mathematical concepts, you can consult a mathematics textbook or online resources such as Khan Academy or Mathway.
Frequently Asked Questions
- What is the slope of a line?
- How do you determine the slope of a line?
- What is the equation of a line?
- How do you calculate the slope of a line?
Answer to Frequently Asked Questions
- The slope of a line is a measure of how steep it is.
- To determine the slope of a line, you can use the formula m = (y2 - y1) / (x2 - x1) or identify the coefficient of x in the equation of the line.
- The equation of a line is typically written in the form y = mx + b, where m is the slope and b is the y-intercept.
- To calculate the slope of a line, you can use the formula m = (y2 - y1) / (x2 - x1) or identify the coefficient of x in the equation of the line.
Slope of a Line: Frequently Asked Questions =============================================
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 you determine the slope of a line?
A: To determine the slope of a line, you can use the formula m = (y2 - y1) / (x2 - x1) or identify the coefficient of x in the equation of the line. The equation of a line is typically written in the form y = mx + b, where m is the slope and b is the y-intercept.
Q: What is the equation of a line?
A: The equation of a line is typically written in the form y = mx + b, where:
- m is the slope of the line
- x is the independent variable (the input or the horizontal axis)
- y is the dependent variable (the output or the vertical axis)
- b is the y-intercept (the point where the line intersects the y-axis)
Q: How do you calculate the slope of a line?
A: To calculate the slope of a line, you can use the formula m = (y2 - y1) / (x2 - x1). This formula calculates the slope as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line.
Q: What is the difference between slope and y-intercept?
A: 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. The slope is a measure of how steep the line is, while the y-intercept is a measure of where the line starts.
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 rises from left to right, a negative slope indicates that the line falls from left to right, and a zero slope indicates that the line is horizontal.
Q: How do you determine the slope of a horizontal line?
A: The slope of a horizontal line is always zero. This is because the line does not rise or fall, but remains at a constant height.
Q: How do you determine the slope of a vertical line?
A: The slope of a vertical line is undefined. This is because the line does not have a defined rate of change, as it extends infinitely in one direction.
Q: Can the slope of a line be a fraction?
A: Yes, the slope of a line can be a fraction. For example, the equation y = 2/3 - 5x has a slope of -5, which is a fraction.
Q: How do you graph a line with a given slope?
A: To graph a line with a given slope, you can use the slope-intercept form of the equation, which is y = mx + b. You can then plot the y-intercept and use the slope to determine the direction and steepness of the line.
Q: Can the slope of a line be negative?
A: Yes, the slope of a line can be negative. A negative slope indicates that the line falls from left to right.
Q: How do you determine the slope of a line with a negative slope?
A: To determine the slope of a line with a negative slope, you can use the formula m = (y2 - y1) / (x2 - x1). If the result is negative, then the slope is negative.
Q: Can the slope of a line be zero?
A: Yes, the slope of a line can be zero. A zero slope indicates that the line is horizontal.
Q: How do you determine the slope of a line with a zero slope?
A: To determine the slope of a line with a zero slope, you can use the formula m = (y2 - y1) / (x2 - x1). If the result is zero, then the slope is zero.
Q: Can the slope of a line be undefined?
A: Yes, the slope of a line can be undefined. This is because the line does not have a defined rate of change, as it extends infinitely in one direction.
Q: How do you determine the slope of a line with an undefined slope?
A: To determine the slope of a line with an undefined slope, you can use the formula m = (y2 - y1) / (x2 - x1). If the denominator is zero, then the slope is undefined.
Conclusion
In conclusion, the slope of a line is a fundamental concept in mathematics that represents the rate of change of a linear equation. It can be positive, negative, or zero, and can be determined using the formula m = (y2 - y1) / (x2 - x1) or by identifying the coefficient of x in the equation of the line.