Math Problem Statement

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

Solution

To determine if the lines 2y+x=42y + x = 4 and y+x+8=0y + 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+by = mx + b, where mm is the slope.

For the first equation 2y+x=42y + x = 4:

  1. Subtract xx from both sides: 2y=x+42y = -x + 4

  2. Divide by 2 to solve for yy: y=12x+2y = -\frac{1}{2}x + 2 The slope of the first line is m1=12m_1 = -\frac{1}{2}.

For the second equation y+x+8=0y + x + 8 = 0:

  1. Subtract x+8x + 8 from both sides: y=x8y = -x - 8 The slope of the second line is m2=1m_2 = -1.

Step 2: Compare the slopes

  • The slope of the first line is m1=12m_1 = -\frac{1}{2}.
  • The slope of the second line is m2=1m_2 = -1.

Since m1m2m_1 \neq m_2, the lines are not parallel.


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.

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Slopes

Formulas

Slope-intercept form y = mx + b
Slope comparison for parallel lines

Theorems

Parallel Line Theorem

Suitable Grade Level

Grades 8-10