Determining the Slope of the New Line

Oct 13, 2025 by leolavanderia.com.br 15 views

Introduction

In mathematics, the slope-intercept form of a line is a fundamental concept that helps us understand the behavior of lines on a coordinate plane. The slope-intercept form is given by the equation $$y = mx + b$$, where $$m$$ is the slope of the line and $$b$$ is the $$y$$-intercept. In this article, we will explore the concept of slope and $$y$$-intercept and determine which graph represents a line with a slope of $$-\frac{2}{3}$$ and a $$y$$-intercept equal to that of the line $$y=\frac{2}{3}x-2$$.

Understanding Slope and $$y$$-intercept

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 denoted by the letter $$m$$ and is a key component of the slope-intercept form of a line. The $$y$$-intercept, denoted by the letter $$b$$, is the point at which the line intersects the $$y$$-axis.

The Given Line

The given line is represented by the equation $$y=\frac{2}{3}x-2$$. To find the slope and $$y$$-intercept of this line, we can compare it to the slope-intercept form of a line. The slope of the given line is $$\frac{2}{3}$$, and the $$y$$-intercept is $$-2$$.

Finding the $$y$$-intercept of the New Line

Since the new line has the same $$y$$-intercept as the given line, we can conclude that the $$y$$-intercept of the new line is also $$-2$$. This means that the new line intersects the $$y$$-axis at the point $$(0, -2)$$.

Determining the Slope of the New Line

The slope of the new line is given as $$-\frac{2}{3}$$. This means that for every unit of horizontal change, the line will change vertically by $$-\frac{2}{3}$$ units.

Graphing the New Line

To graph the new line, we can use the slope-intercept form of a line. We know that the line intersects the $$y$$-axis at the point $$(0, -2)$$ and has a slope of $$-\frac{2}{3}$$. We can use this information to plot the line on a coordinate plane.

Comparing the Graphs

We are given four graphs to compare. We need to determine which graph represents the line with a slope of $$-\frac{2}{3}$$ and a $$y$$-intercept equal to that of the line $$y=\frac{2}{3}x-2$$. To do this, we can compare the slope and $$y$$-intercept of each graph to the given information.

Graph 1

Graph 1 has a slope of $$\frac{2}{3}$$ and a $$y$$-intercept of $$-2$$. This means that the line intersects the $$y$$-axis at the point $$(0, -2)$$ and has a slope of $$\frac{2}{3}$$. However, the slope of the new line is $$-\frac{2}{3}$$, so this graph does not represent the new line.

Graph 2

Graph 2 has a slope of $$-\frac{2}{3}$$ and a $$y$$-intercept of $$-2$$. This means that the line intersects the $$y$$-axis at the point $$(0, -2)$$ and has a slope of $$-\frac{2}{3}$$. This matches the given information, so this graph represents the new line.

Graph 3

Graph 3 has a slope of $$\frac{2}{3}$$ and a $$y$$-intercept of $$2$$. This means that the line intersects the $$y$$-axis at the point $$(0, 2)$$ and has a slope of $$\frac{2}{3}$$. However, the slope of the new line is $$-\frac{2}{3}$$, and the $$y$$-intercept is $$-2$$, so this graph does not represent the new line.

Graph 4

Graph 4 has a slope of $$-\frac{2}{3}$$ and a $$y$$-intercept of $$2$$. This means that the line intersects the $$y$$-axis at the point $$(0, 2)$$ and has a slope of $$-\frac{2}{3}$$. However, the $$y$$-intercept of the new line is $$-2$$, so this graph does not represent the new line.

Conclusion

In conclusion, the graph that represents a line with a slope of $$-\frac{2}{3}$$ and a $$y$$-intercept equal to that of the line $$y=\frac{2}{3}x-2$$ is Graph 2. This graph intersects the $$y$$-axis at the point $$(0, -2)$$ and has a slope of $$-\frac{2}{3}$$. This matches the given information, and we can conclude that Graph 2 is the correct answer.

Final Answer

The final answer is Graph 2.

Introduction

In our previous article, we explored the concept of slope and $$y$$-intercept and determined which graph represents a line with a slope of $$-\frac{2}{3}$$ and a $$y$$-intercept equal to that of the line $$y=\frac{2}{3}x-2$$. In this article, we will answer some frequently asked questions about slope and $$y$$-intercept to help you better understand these concepts.

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: $$m = \frac{y_2 - y_1}{x_2 - x_1}$$, where $$(x_1, y_1)$$ and $$(x_2, y_2)$$ are two points on the line.

Q3: What is the $$y$$-intercept of a line?

A3: The $$y$$-intercept of a line is the point at which the line intersects the $$y$$-axis. It is denoted by the letter $$b$$ in the slope-intercept form of a line.

Q4: How do I find the $$y$$-intercept of a line?

A4: To find the $$y$$-intercept of a line, you can set $$x = 0$$ in the equation of the line and solve for $$y$$.

Q5: What is the difference between a positive and negative slope?

A5: A positive slope indicates that the line is rising from left to right, while a negative slope indicates that the line is falling from left to right.

Q6: How do I graph a line with a given slope and $$y$$-intercept?

A6: To graph a line with a given slope and $$y$$-intercept, you can use the slope-intercept form of a line. You can plot the $$y$$-intercept on the coordinate plane and then use the slope to determine the direction of the line.

Q7: What is the equation of a line with a slope of $$-\frac{2}{3}$$ and a $$y$$-intercept of $$-2$$?

A7: The equation of a line with a slope of $$-\frac{2}{3}$$ and a $$y$$-intercept of $$-2$$ is $$y = -\frac{2}{3}x - 2$$.

Q8: How do I determine which graph represents a line with a given slope and $$y$$-intercept?

A8: To determine which graph represents a line with a given slope and $$y$$-intercept, you can compare the slope and $$y$$-intercept of each graph to the given information.

Q9: What is the relationship between the slope and $$y$$-intercept of a line?

A9: The slope and $$y$$-intercept of a line are related in that the slope determines the direction of the line, while the $$y$$-intercept determines the point at which the line intersects the $$y$$-axis.

Q10: How do I use the slope-intercept form of a line to solve problems?

A10: To use the slope-intercept form of a line to solve problems, you can substitute the given values into the equation and solve for the unknown variable.

Conclusion

In conclusion, we have answered some frequently asked questions about slope and $$y$$-intercept to help you better understand these concepts. We hope that this article has been helpful in clarifying any confusion you may have had about slope and $$y$$-intercept.

Final Answer

The final answer is that the slope and $$y$$-intercept are two important components of the slope-intercept form of a line, and understanding these concepts is crucial for solving problems in mathematics.