The Equations Of Three Lines Are Given Below:Line 1: $5y = 2x + 4$Line 2: $y = \frac{2}{5}x - 6$Line 3: $10x - 4y = 6$For Each Pair Of Lines, Determine Whether They Are Parallel, Perpendicular, Or Neither.1. Line 1 And Line

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The Equations of Three Lines: Parallel, Perpendicular, or Neither?

In mathematics, the study of lines and their relationships is a fundamental concept in geometry and algebra. When given the equations of two or more lines, it is essential to determine their relationship, whether they are parallel, perpendicular, or neither. In this article, we will explore the equations of three lines and determine their relationships for each pair.

Line 1: 5y=2x+45y = 2x + 4

The equation of Line 1 is given as 5y=2x+45y = 2x + 4. To determine its slope, we can rewrite the equation in the slope-intercept form, y=mx+by = mx + b, where mm is the slope and bb is the y-intercept.

# Import necessary modules
import sympy as sp

# Define variables
x, y = sp.symbols('x y')

# Define the equation of Line 1
eq1 = sp.Eq(5*y, 2*x + 4)

# Solve for y
sol = sp.solve(eq1, y)

# Print the equation in slope-intercept form
print(f"y = {sol[0]}")

The equation of Line 1 in slope-intercept form is y=25x+45y = \frac{2}{5}x + \frac{4}{5}. Therefore, the slope of Line 1 is 25\frac{2}{5}.

Line 2: y=25x−6y = \frac{2}{5}x - 6

The equation of Line 2 is given as y=25x−6y = \frac{2}{5}x - 6. We can see that the slope of Line 2 is also 25\frac{2}{5}, which is the same as the slope of Line 1.

Line 3: 10x−4y=610x - 4y = 6

The equation of Line 3 is given as 10x−4y=610x - 4y = 6. To determine its slope, we can rewrite the equation in the slope-intercept form.

# Define the equation of Line 3
eq3 = sp.Eq(10*x - 4*y, 6)

# Solve for y
sol = sp.solve(eq3, y)

# Print the equation in slope-intercept form
print(f"y = {sol[0]}")

The equation of Line 3 in slope-intercept form is y=52x−32y = \frac{5}{2}x - \frac{3}{2}. Therefore, the slope of Line 3 is 52\frac{5}{2}.

Relationship between Line 1 and Line 2

Since the slopes of Line 1 and Line 2 are the same, we can conclude that they are parallel.

Relationship between Line 1 and Line 3

Since the slopes of Line 1 and Line 3 are different, we can conclude that they are neither parallel nor perpendicular.

Relationship between Line 2 and Line 3

Since the slopes of Line 2 and Line 3 are different, we can conclude that they are neither parallel nor perpendicular.

In conclusion, we have determined the relationships between the three lines. Line 1 and Line 2 are parallel, while Line 1 and Line 3, and Line 2 and Line 3 are neither parallel nor perpendicular.

The study of lines and their relationships is a fundamental concept in mathematics. By understanding the equations of lines and their slopes, we can determine their relationships and make conclusions about their behavior. This knowledge is essential in various fields, including geometry, algebra, and engineering.

The following is a Python code snippet that calculates the slope of a line given its equation.

import sympy as sp

def calculate_slope(eq):
    x, y = sp.symbols('x y')
    sol = sp.solve(eq, y)
    slope = sp.diff(sol[0], x)
    return slope

# Test the function
eq = sp.Eq(5*y, 2*x + 4)
print(calculate_slope(eq))
```<br/>
**The Equations of Three Lines: Parallel, Perpendicular, or Neither? - Q&A**

**Introduction**
===============

In our previous article, we explored the equations of three lines and determined their relationships for each pair. In this article, we will answer some frequently asked questions (FAQs) related to the topic.

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

A: 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).

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

A: To determine if two lines are parallel or perpendicular, you can compare their slopes. If the slopes are equal, the lines are parallel. If the slopes are negative reciprocals of each other (i.e., one slope is the negative reciprocal of the other), the lines are perpendicular.

**Q: What is the slope of a line?**
------------------------------

A: The slope of a line is a measure of how steep the line 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 can I calculate the slope of a line?**
------------------------------------------------

A: To calculate the slope of a line, you can use the following formula:

m = (y2 - y1) / (x2 - x1)

where m is the slope, and (x1, y1) and (x2, y2) are two points on the line.

**Q: What is the equation of a line?**
-----------------------------------

A: The equation of a line is a mathematical expression that describes the relationship between the x and y coordinates of points on the line. It is typically written in the form:

y = mx + b

where m is the slope, and b is the y-intercept.

**Q: How can I write the equation of a line in slope-intercept form?**
----------------------------------------------------------------

A: To write the equation of a line in slope-intercept form, you can use the following formula:

y = mx + b

where m is the slope, and b is the y-intercept.

**Q: What is the y-intercept of a line?**
--------------------------------------

A: The y-intercept of a line is the point where the line intersects the y-axis. It is the value of y when x is equal to zero.

**Q: How can I find the y-intercept of a line?**
------------------------------------------------

A: To find the y-intercept of a line, you can set x equal to zero in the equation of the line and solve for y.

**Q: What is the relationship between the slope and the y-intercept of a line?**
-------------------------------------------------------------------------

A: The slope and the y-intercept of a line are related in that the slope determines the steepness of the line, while the y-intercept determines the position of the line on the y-axis.

**Q: How can I use the slope and y-intercept to graph a line?**
---------------------------------------------------------

A: To graph a line, you can use the slope and y-intercept to find two points on the line and then draw a line through those points.

**Conclusion**
=============

In conclusion, we have answered some frequently asked questions related to the equations of three lines and their relationships. We hope this article has been helpful in clarifying any doubts you may have had.

**Final Thoughts**
================

The study of lines and their relationships is a fundamental concept in mathematics. By understanding the equations of lines and their slopes, we can determine their relationships and make conclusions about their behavior. This knowledge is essential in various fields, including geometry, algebra, and engineering.

**References**
==============

* [1] Sympy Documentation. (n.d.). Retrieved from <https://docs.sympy.org/latest/index.html>
* [2] Khan Academy. (n.d.). Retrieved from <https://www.khanacademy.org/math>

**Appendix**
==========

The following is a Python code snippet that calculates the slope and y-intercept of a line given its equation.

```python
import sympy as sp

def calculate_slope(eq):
    x, y = sp.symbols('x y')
    sol = sp.solve(eq, y)
    slope = sp.diff(sol[0], x)
    return slope

def calculate_y_intercept(eq):
    x, y = sp.symbols('x y')
    sol = sp.solve(eq, y)
    y_intercept = sol[0].subs(x, 0)
    return y_intercept

# Test the functions
eq = sp.Eq(5*y, 2*x + 4)
print(calculate_slope(eq))
print(calculate_y_intercept(eq))