What are parallel lines?

Parallel lines are lines that never touch, no matter how far they extend.

Think of:

• Railroad tracks

• The edges of a ruler

• Lines on notebook paper

They always stay the same distance apart.

Key property of parallel lines

Parallel lines have the same slope.

• If two lines are parallel:

• The y-intercept can be different:

• Same slope (2), different y-intercepts → never touch

Example 1: Identifying parallel lines

Line 1:

Line 2:

• Both slopes = 3 → parallel

• Different y-intercepts → do not overlap

Example 2: Finding an equation of a line parallel to another

Given:
Line: y = −1/2x + 4
Point: (2, 1)

Step 1: Use the same slope

Step 2: Use point-slope form

Step 3: Simplify (optional)

✅ This line is parallel to the original.

Important things to remember

1. Slope equality is key

   • Parallel → slopes are equal

   • Intercepts can be different

2. Never intersect

   • No matter how far the lines go

3. Graphing tip

   • Start with slope-intercept form (y = mx + b)

   • Make sure slopes match to be parallel

Visual summary

Common beginner mistakes

• ❌ Thinking lines with same y-intercept are always parallel
  ✅ They’re the same line if both slope and intercept are identical

• ❌ Forgetting slope must match exactly
  ✅ Even a small difference → lines are not parallel

• ❌ Mixing up horizontal and vertical lines
  ✅ Horizontal lines: y = constant (parallel if same y)
  ✅ Vertical lines: x = constant (parallel if same x)

Why parallel lines matter

• Used in geometry (angles, shapes)

• Helps solve linear systems

• Important for real-world applications: roads, construction, designs