Given Line and Its Properties

Oct 13, 2025 by magicalceylontours.com 72 views

Introduction

In mathematics, particularly in algebra and geometry, understanding the properties of lines is crucial for solving various problems. One of the key properties of a line is its slope, which represents the rate of change of the line. In this article, we will explore how to determine the graph of a line with a given slope and $y$ -intercept.

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 calculated as:

m = \frac{\text{rise}}{\text{run}}

The $y$ -intercept of a line is the point where the line intersects the $y$ -axis. It is denoted by the letter $b$ and is the value of $y$ when $x=0$ .

Given Line and Its Properties

The given line is $y=\frac{2}{3}x - 2$ . We can determine its slope and $y$ -intercept as follows:

Slope ( $m$ ): $\frac{2}{3}$
$y$ -intercept ( $b$ ): $-2$

Line with a Slope of $-\frac{2}{3}$ and a $y$ -intercept Equal to that of the Given Line

We are asked to find the graph of a line with a slope of $-\frac{2}{3}$ and a $y$ -intercept equal to that of the given line. To do this, we need to find the equation of the line.

Equation of the Line

The equation of a line with a slope of $m$ and a $y$ -intercept of $b$ is given by:

y = mx + b

In this case, the slope is $-\frac{2}{3}$ and the $y$ -intercept is $-2$ . Therefore, the equation of the line is:

y = -\frac{2}{3}x - 2

Graph of the Line

To graph the line, we can use the equation $y = -\frac{2}{3}x - 2$ . We can start by plotting the $y$ -intercept, which is $-2$ . Then, we can use the slope to find another point on the line.

Finding Another Point on the Line

To find another point on the line, we can use the slope to find the $x$ -coordinate of the point. Since the slope is $-\frac{2}{3}$ , we can choose an $x$ -value and calculate the corresponding $y$ -value.

Let's choose $x=3$ . Then, the corresponding $y$ -value is:

y = -\frac{2}{3}(3) - 2 = -4

Therefore, the point $(3, -4)$ is on the line.

Plotting the Line

Now that we have two points on the line, we can plot the line. We can use a graphing tool or draw the line by hand.

Conclusion

In this article, we explored how to determine the graph of a line with a given slope and $y$ -intercept. We used the equation of a line to find the equation of the line with a slope of $-\frac{2}{3}$ and a $y$ -intercept equal to that of the given line. We then plotted the line using the equation and two points on the line.

Key Takeaways

The slope of a line is a measure of how steep it is.
The $y$ -intercept of a line is the point where the line intersects the $y$ -axis.
The equation of a line with a slope of $m$ and a $y$ -intercept of $b$ is given by $y = mx + b$ .
To graph a line, we can use the equation of the line and plot two points on the line.

Final Answer

Frequently Asked Questions

Q: What is the slope of the line $y=\frac{2}{3}x - 2$ ?

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

Q: What is the $y$ -intercept of the line $y=\frac{2}{3}x - 2$ ?

A: The $y$ -intercept of the line $y=\frac{2}{3}x - 2$ is $-2$ .

Q: What is the equation of the line with a slope of $-\frac{2}{3}$ and a $y$ -intercept equal to that of the line $y=\frac{2}{3}x - 2$ ?

A: The equation of the line with a slope of $-\frac{2}{3}$ and a $y$ -intercept equal to that of the line $y=\frac{2}{3}x - 2$ is $y = -\frac{2}{3}x - 2$ .

Q: How do I graph the line with a slope of $-\frac{2}{3}$ and a $y$ -intercept equal to that of the line $y=\frac{2}{3}x - 2$ ?

A: To graph the line with a slope of $-\frac{2}{3}$ and a $y$ -intercept equal to that of the line $y=\frac{2}{3}x - 2$ , you can use the equation $y = -\frac{2}{3}x - 2$ and plot two points on the line.

Q: What are the key takeaways from this article?

A: The key takeaways from this article are:

The slope of a line is a measure of how steep it is.
The $y$ -intercept of a line is the point where the line intersects the $y$ -axis.
The equation of a line with a slope of $m$ and a $y$ -intercept of $b$ is given by $y = mx + b$ .
To graph a line, you can use the equation of the line and plot two points on the line.

Q: What is the final answer to the problem?

A: The final answer to the problem is the graph of the line with a slope of $-\frac{2}{3}$ and a $y$ -intercept equal to that of the line $y=\frac{2}{3}x - 2$ , which is the line with the equation $y = -\frac{2}{3}x - 2$ .

Additional Resources

For more information on lines and their properties, see the article on "Lines and Their Properties".
For more information on graphing lines, see the article on "Graphing Lines".
For more information on the equation of a line, see the article on "The Equation of a Line".

Conclusion

In this article, we answered frequently asked questions about lines with a slope of $-\frac{2}{3}$ and a $y$ -intercept equal to that of the line $y=\frac{2}{3}x - 2$ . We provided key takeaways and a final answer to the problem. We also provided additional resources for further learning.

Final Answer

The final answer to the problem is the graph of the line with a slope of $-\frac{2}{3}$ and a $y$ -intercept equal to that of the line $y=\frac{2}{3}x - 2$ , which is the line with the equation $y = -\frac{2}{3}x - 2$ .