Which Graph Represents A Line With A Slope Of $-\frac{2}{3}$ And A Y Y Y -intercept Equal To That Of The Line Y = 2 3 X − 2 Y=\frac{2}{3}x-2 Y = 3 2 ​ X − 2 ?

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Introduction

In mathematics, a line is a set of points that extend infinitely in two directions. It is often represented graphically on a coordinate plane, with the xx-axis representing the horizontal direction and the yy-axis representing the vertical direction. The slope of a line is a measure of how steep it is, and it is calculated as the ratio of the vertical change (the rise) to the horizontal change (the run). The yy-intercept of a line is the point at which it intersects the yy-axis.

In this article, we will explore the concept of a line with a slope of $-\frac{2}{3}$ and a yy-intercept equal to that of the line y=23x−2y=\frac{2}{3}x-2. We will examine the characteristics of this line and determine which graph represents it.

Characteristics of the Line

The slope of a line is a key characteristic that determines its steepness. A negative slope indicates that the line slopes downward from left to right, while a positive slope indicates that the line slopes upward from left to right. In this case, the slope of the line is $-\frac{2}{3}$, which is negative.

The yy-intercept of a line is the point at which it intersects the yy-axis. It is the value of yy when x=0x=0. In this case, the yy-intercept of the line is equal to that of the line y=23x−2y=\frac{2}{3}x-2. To find the yy-intercept of this line, we can substitute x=0x=0 into the equation:

y=23(0)−2=−2y=\frac{2}{3}(0)-2=-2

Therefore, the yy-intercept of the line is −2-2.

Graphing the Line

To graph the line, we can use the slope-intercept form of a linear equation, which is y=mx+by=mx+b, where mm is the slope and bb is the yy-intercept. In this case, the slope is $-\frac{2}{3}$ and the yy-intercept is −2-2. Therefore, the equation of the line is:

y=−23x−2y=-\frac{2}{3}x-2

To graph the line, we can start by plotting the yy-intercept, which is the point (0,−2)(0,-2). Then, we can use the slope to determine the direction of the line. Since the slope is negative, the line will slope downward from left to right.

Determining the Correct Graph

We are given four graphs to choose from, and we need to determine which one represents the line with a slope of $-\frac{2}{3}$ and a yy-intercept equal to that of the line y=23x−2y=\frac{2}{3}x-2. Let's examine each graph carefully.

Graph A

Graph A has a slope of $-\frac{2}{3}$ and a yy-intercept of −2-2. It is the correct graph.

Graph B

Graph B has a slope of $\frac{2}{3}$ and a yy-intercept of −2-2. It is not the correct graph because the slope is positive, not negative.

Graph C

Graph C has a slope of $-\frac{2}{3}$ and a yy-intercept of −1-1. It is not the correct graph because the yy-intercept is not equal to that of the line y=23x−2y=\frac{2}{3}x-2.

Graph D

Graph D has a slope of $-\frac{2}{3}$ and a yy-intercept of −3-3. It is not the correct graph because the yy-intercept is not equal to that of the line y=23x−2y=\frac{2}{3}x-2.

Conclusion

In conclusion, the graph that represents a line with a slope of $-\frac{2}{3}$ and a yy-intercept equal to that of the line y=23x−2y=\frac{2}{3}x-2 is Graph A. This graph has a slope of $-\frac{2}{3}$ and a yy-intercept of −2-2, which matches the characteristics of the line.

Frequently Asked Questions

  • What is the slope of the line? The slope of the line is $-\frac{2}{3}$.
  • What is the yy-intercept of the line? The yy-intercept of the line is −2-2.
  • Which graph represents the line? Graph A represents the line.

Final Answer

The final answer is Graph A.

Introduction

In our previous article, we explored the concept of a line with a slope of $-\frac{2}{3}$ and a yy-intercept equal to that of the line y=23x−2y=\frac{2}{3}x-2. We examined the characteristics of this line and determined which graph represents it. In this article, we will answer some frequently asked questions related to this topic.

Q&A

Q: What is the slope of the line?

A: The slope of the line is $-\frac{2}{3}$.

Q: What is the yy-intercept of the line?

A: The yy-intercept of the line is −2-2.

Q: Which graph represents the line?

A: Graph A represents the line.

Q: Why is Graph A the correct graph?

A: Graph A is the correct graph because it has a slope of $-\frac{2}{3}$ and a yy-intercept of −2-2, which matches the characteristics of the line.

Q: What is the equation of the line?

A: The equation of the line is y=−23x−2y=-\frac{2}{3}x-2.

Q: How do I graph the line?

A: To graph the line, start by plotting the yy-intercept, which is the point (0,−2)(0,-2). Then, use the slope to determine the direction of the line. Since the slope is negative, the line will slope downward from left to right.

Q: What if I have a different slope or yy-intercept?

A: If you have a different slope or yy-intercept, you will need to adjust the equation of the line accordingly. For example, if you have a slope of $\frac{2}{3}$ and a yy-intercept of −1-1, the equation of the line would be y=23x−1y=\frac{2}{3}x-1.

Q: Can I use a graphing calculator to graph the line?

A: Yes, you can use a graphing calculator to graph the line. Simply enter the equation of the line into the calculator and press the graph button.

Conclusion

In conclusion, we have answered some frequently asked questions related to the topic of a line with a slope of $-\frac{2}{3}$ and a yy-intercept equal to that of the line y=23x−2y=\frac{2}{3}x-2. We hope that this article has been helpful in clarifying any confusion and providing a better understanding of this topic.

Final Answer

The final answer is Graph A.