Math Problem Statement

solve

Solution

To determine whether the given lines are parallel, perpendicular, or neither, we need to analyze their slopes. The equations of the lines given are:

y=x+3y = x + 3 y=2x+3y = 2x + 3

These equations are in the slope-intercept form, y=mx+by = mx + b, where mm is the slope and bb is the y-intercept.

  • For the first equation y=x+3y = x + 3, the slope m1=1m_1 = 1.
  • For the second equation y=2x+3y = 2x + 3, the slope m2=2m_2 = 2.

Analysis:

  1. Parallel Lines: Two lines are parallel if their slopes are equal. Here, m1=1m_1 = 1 and m2=2m_2 = 2, so the slopes are not equal. Therefore, the lines are not parallel.

  2. Perpendicular Lines: Two lines are perpendicular if the product of their slopes is 1-1. Let's check: m1×m2=1×2=2m_1 \times m_2 = 1 \times 2 = 2 Since 212 \neq -1, the lines are not perpendicular.

Conclusion:

The given lines are neither parallel nor perpendicular.

Would you like more details or have any other questions?

Here are some related questions you might consider:

  1. What are the characteristics of parallel lines?
  2. How do you find the slope from an equation not in slope-intercept form?
  3. Can lines with the same y-intercept ever be perpendicular?
  4. How would the problem change if one of the slopes were negative?
  5. What does the y-intercept tell us about a line?

Tip: Always check both slope and y-intercept when comparing lines. Even if they have the same y-intercept, their slopes determine if they are parallel or perpendicular.

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Math Problem Analysis

Mathematical Concepts

Slope
Parallel Lines
Perpendicular Lines

Formulas

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Theorems

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Suitable Grade Level

Grades 9-12