https://file.aiquickdraw.com/mathbot/file/17285805403381c7o3iil.jpg

The image contains a set of linear equations, along with a question in Persian. Here’s a breakdown of what I can interpret from the equations and the problem:

### Equations:
1. \( 2x + 4y + 1 = 0 \)
2. \( 5x - y + 1 = 0 \)

### Question:
In Persian, the problem asks you to do the following:
- (الف) Show that these two lines are **parallel**.
- (ب) Calculate the **distance** between these two lines.

### Step-by-Step Solution:

#### (الف) Showing the lines are parallel:

The general form of a line is \(Ax + By + C = 0\), where the slope \(m\) is given by:
\[
m = -\frac{A}{B}
\]
For the first line \(2x + 4y + 1 = 0\):
\[
m_1 = -\frac{2}{4} = -\frac{1}{2}
\]
For the second line \(5x - y + 1 = 0\) (which can be rewritten as \(5x + (-1)y + 1 = 0\)):
\[
m_2 = -\frac{5}{-1} = 5
\]
Thus, the slopes of the two lines are not equal. This means **the lines are not parallel**. 

If the question intends to ask for an error check, it's worth rechecking if we interpreted the two original equations correctly.

#### (ب) Distance Between the Two Lines:
If they were meant to be parallel, the formula for the distance between two parallel lines of the form \(Ax + By + C_1 = 0\) and \(Ax + By + C_2 = 0\) is:
\[
\text{Distance} = \frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}}
\]
However, this formula only applies if the lines are parallel, which they are not based on the slopes calculated earlier.

Would you like me to clarify any further details, or solve for any other scenario? Let me know if you'd like additional steps explained.

### 5 Follow-up Questions:
1. How do you calculate the distance between two non-parallel lines?
2. What is the significance of finding the slope of a line?
3. Can two lines be coincident if their slopes are equal?
4. How can you transform equations into the standard slope-intercept form \(y = mx + b\)?
5. Why is the distance between two lines important in geometry?

### Tip:
When working with systems of linear equations, always check the slopes first to identify if the lines are parallel, intersecting, or coincident.

Show that the given lines are parallel and calculate the distance between them.

Learn how to determine if two lines are parallel using their slopes and calculate the distance between them with detailed step-by-step instructions. This problem covers key algebraic concepts related to linear equations and slopes, ideal for students in Grades 9-11.

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