Understanding Parallel and Perpendicular Lines

Oct 13, 2025 by miodeya.com 39 views

What Statements are True About the Two Equations?

In mathematics, equations are used to represent relationships between variables. When dealing with linear equations, it's essential to understand the properties of parallel and perpendicular lines. In this article, we will explore the relationship between two given equations and determine which statements are true about them.

Understanding Parallel and Perpendicular Lines

Before we dive into the given equations, let's briefly discuss parallel and perpendicular lines. Parallel lines are lines that lie in the same plane and never intersect, regardless of how far they are extended. Perpendicular lines, on the other hand, are lines that intersect at a right angle (90 degrees).

The Given Equations

The two given equations are:

${ \begin{align*} y &= -18x - 42 \\ -2x - \frac{1}{9}y &= 13 \end{align*} }$

To determine the relationship between these equations, we need to rewrite them in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.

Rewriting the Equations

Let's rewrite the first equation in slope-intercept form:

${ y = -18x - 42 }$

This equation is already in slope-intercept form, with a slope of -18 and a y-intercept of -42.

Now, let's rewrite the second equation in slope-intercept form:

${ -2x - \frac{1}{9}y = 13 }$

To isolate y, we can multiply both sides of the equation by -9:

${ -18x - y = -117 }$

Now, we can rewrite the equation as:

${ y = -18x + 117 }$

This equation is also in slope-intercept form, with a slope of -18 and a y-intercept of 117.

Comparing the Equations

Now that we have rewritten both equations in slope-intercept form, we can compare their slopes. If the slopes are equal, the lines are parallel. If the slopes are negative reciprocals of each other, the lines are perpendicular.

The Slopes of the Equations

The slope of the first equation is -18, and the slope of the second equation is also -18. Since the slopes are equal, the lines are parallel.

Conclusion

Based on our analysis, we can conclude that the two given equations represent parallel lines. Therefore, statement A is true, and statements B and C are false.

Final Answer

The final answer is:

A. The equations are parallel lines.

Discussion

This problem requires a good understanding of linear equations and the properties of parallel and perpendicular lines. By rewriting the equations in slope-intercept form and comparing their slopes, we can determine the relationship between the lines.

Additional Tips

When dealing with linear equations, it's essential to rewrite them in slope-intercept form to easily compare their slopes.
Parallel lines have equal slopes, while perpendicular lines have negative reciprocal slopes.
By understanding the properties of parallel and perpendicular lines, we can solve problems involving linear equations and graphing.
Q&A: Understanding Parallel and Perpendicular Lines

In our previous article, we explored the relationship between two given equations and determined that they represent parallel lines. In this article, we will answer some frequently asked questions about parallel and perpendicular lines.

Q: What are parallel lines?

A: Parallel lines are lines that lie in the same plane and never intersect, regardless of how far they are extended. They have the same slope and never touch each other.

Q: What are perpendicular lines?

A: Perpendicular lines are lines that intersect at a right angle (90 degrees). They have negative reciprocal slopes, meaning that if one line has a slope of m, the other line has a slope of -1/m.

Q: How can I determine if two lines are parallel or perpendicular?

A: To determine if two lines are parallel or perpendicular, you can rewrite them in slope-intercept form (y = mx + b) and compare their slopes. If the slopes are equal, the lines are parallel. If the slopes are negative reciprocals of each other, the lines are perpendicular.

Q: What is the difference between parallel and perpendicular lines?

A: The main difference between parallel and perpendicular lines is their slope. Parallel lines have the same slope, while perpendicular lines have negative reciprocal slopes.

Q: Can parallel lines intersect?

A: No, parallel lines cannot intersect. They lie in the same plane and never touch each other.

Q: Can perpendicular lines be parallel?

A: No, perpendicular lines cannot be parallel. Perpendicular lines intersect at a right angle, while parallel lines never intersect.

Q: How can I graph parallel and perpendicular lines?

A: To graph parallel and perpendicular lines, you can use the slope-intercept form (y = mx + b) and plot points on a coordinate plane. For parallel lines, use the same slope and different y-intercepts. For perpendicular lines, use negative reciprocal slopes and different y-intercepts.

Q: What are some real-world examples of parallel and perpendicular lines?

A: Some real-world examples of parallel and perpendicular lines include:

Railway tracks: Parallel lines that never intersect.
Building walls: Perpendicular lines that intersect at a right angle.
Grid lines on a map: Parallel and perpendicular lines that help with navigation.

Conclusion

In this article, we answered some frequently asked questions about parallel and perpendicular lines. By understanding the properties of these lines, we can solve problems involving linear equations and graphing.

Final Tips

Always rewrite linear equations in slope-intercept form to easily compare their slopes.
Parallel lines have equal slopes, while perpendicular lines have negative reciprocal slopes.
By understanding the properties of parallel and perpendicular lines, we can solve problems involving linear equations and graphing.

Additional Resources

For more information on linear equations and graphing, check out our previous article: What Statements are True About the Two Equations?
For more practice problems and examples, try the following resources:
- Khan Academy: Linear Equations and Graphing
- Mathway: Linear Equations and Graphing
- IXL: Linear Equations and Graphing