Are the lines 2y+x=4 and y+x+8=0 parallel

To determine if the lines \( 2y + x = 4 \) and \( y + x + 8 = 0 \) are parallel, we need to find their slopes. Parallel lines have the same slope.

### Step 1: Rewrite each equation in slope-intercept form \( y = mx + b \), where \( m \) is the slope.

#### For the first equation \( 2y + x = 4 \):

1. Subtract \( x \) from both sides:
   \[
   2y = -x + 4
   \]

2. Divide by 2 to solve for \( y \):
   \[
   y = -\frac{1}{2}x + 2
   \]
   The slope of the first line is \( m_1 = -\frac{1}{2} \).

#### For the second equation \( y + x + 8 = 0 \):

1. Subtract \( x + 8 \) from both sides:
   \[
   y = -x - 8
   \]
   The slope of the second line is \( m_2 = -1 \).

### Step 2: Compare the slopes

- The slope of the first line is \( m_1 = -\frac{1}{2} \).
- The slope of the second line is \( m_2 = -1 \).

Since \( m_1 \neq m_2 \), the lines are **not parallel**.

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Would you like more details on how to determine slopes or solve linear equations? Here are some related questions you might find useful:

1. What are the conditions for two lines to be perpendicular?
2. How do you find the point of intersection between two lines?
3. Can two lines have the same slope but not be parallel?
4. What is the general equation of a line in 3D space?
5. How can you check if a set of lines is concurrent?

**Tip:** When comparing the slopes of two lines, remember that parallel lines will always have identical slopes, while perpendicular lines have slopes that are negative reciprocals of each other.

Are the lines 2y + x = 4 and y + x + 8 = 0 parallel?

Explore how to determine if the lines 2y + x = 4 and y + x + 8 = 0 are parallel by comparing their slopes. This step-by-step solution covers key concepts in algebra, including slope-intercept form and the Parallel Line Theorem. Suitable for students in Grades 8-10.

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