Lesson 8: Parallel Lines
Learning Objectives
By the end of this lesson, students will be able to:
- Identify parallel lines from their equations
- Determine if lines are parallel using gradients
- Find equations of parallel lines
Parallel Lines
Two lines are parallel if they have the same gradient and never intersect.
If two lines are parallel, then \(m_1 = m_2\)
Example 1: Identifying Parallel Lines
Are the lines \(y = 2x + 3\) and \(y = 2x - 5\) parallel?
Solution: Both lines have gradient \(m = 2\), so they are parallel.
Example 2: Finding Parallel Line Equation
Find the equation of the line parallel to \(y = 3x + 1\) that passes through the point (2, 7).
Solution: - Parallel lines have the same gradient: \(m = 3\) - Using point-slope form with (2, 7): \[y - 7 = 3(x - 2)\] \[y - 7 = 3x - 6\] \[y = 3x + 1\]
Verification: When \(x = 2\), \(y = 3(2) + 1 = 7\) ✓
Practice Exercises
Determine if these pairs of lines are parallel or not:
- \(y = 3x + 2\) and \(y = 3x - 7\)
- \(y = 2x + 1\) and \(y = -\frac{1}{2}x + 4\)
- \(y = 5x\) and \(y = -5x + 3\)
Find the equation of the line parallel to \(y = -2x + 3\) through point (1, 5).
The line \(L_1\) passes through points A(1, 2) and B(5, 10). Find the equation of the line parallel to \(L_1\) through C(0, 0).
Worksheet 3A: Parallel Lines Practice
- Determine whether each pair of lines is parallel or not:
- \(y = 4x + 1\) and \(y = 4x - 3\)
- \(y = 2x + 5\) and \(y = -\frac{1}{2}x + 2\)
- Find the equation of:
- The line parallel to \(y = 5x + 2\) through point (2, 3)
- The line parallel to \(2x + y - 4 = 0\) through point (-1, 5)
Problem-Solving Strategy
When working with parallel lines:
- Identify or calculate the gradient of the given line
- Use the same gradient for the parallel line
- Use the point-slope form with the given point
- Simplify to the required form
Real-World Applications
- Engineering: Designing parallel roads or railway tracks
- Architecture: Laying out evenly spaced features
Homework
Complete the parallel lines practice set.
- Parallel, not parallel
- \(y = 5x - 7\), \(y = -2x + 3\)
Previous Lesson: Lesson 7: Linear Equations and Intersections
Next Lesson: Lesson 9: Perpendicular Lines