Parallel and Perpendicular Lines: Finding Equations

In geometry, understanding the relationships between lines is fundamental. Two common relationships are parallelism and perpendicularity. Parallel lines never intersect, while perpendicular lines intersect at a right angle. This lesson explores how to find the equations of lines that are parallel or perpendicular to a given line and pass through a specific point.

Parallel Lines

Parallel lines have the same slope. This means that if you have the equation of one line, you can easily find the equation of a parallel line by keeping the slope the same and adjusting the y-intercept.

Example 1:

Find the equation of the line that is parallel to the line y = 2x + 3 and passes through the point (1, 5).

**Solution:**

1. The given line has a slope of 2 (the coefficient of x).
2. Since we want a parallel line, the new line will also have a slope of 2.
3. We can use the point-slope form of a linear equation to find the equation of the new line:

y – y1 = m(x – x1)

where m is the slope and (x1, y1) is the given point.

Substituting the values, we get:

y – 5 = 2(x – 1)

Simplifying the equation, we get:

y = 2x + 3

Therefore, the equation of the line parallel to y = 2x + 3 and passing through (1, 5) is y = 2x + 3.

Perpendicular Lines

Perpendicular lines have slopes that are negative reciprocals of each other. This means that if the slope of one line is m, the slope of a perpendicular line is -1/m.

Example 2:

Find the equation of the line that is perpendicular to the line y = -3x + 1 and passes through the point (2, 4).

**Solution:**

1. The given line has a slope of -3.
2. The slope of a perpendicular line is the negative reciprocal of -3, which is 1/3.
3. Using the point-slope form, we get:

y – 4 = (1/3)(x – 2)

Simplifying, we get:

y = (1/3)x + 10/3

Therefore, the equation of the line perpendicular to y = -3x + 1 and passing through (2, 4) is y = (1/3)x + 10/3.

Key Takeaways

• Parallel lines have the same slope.
• Perpendicular lines have slopes that are negative reciprocals of each other.
• The point-slope form of a linear equation (y – y1 = m(x – x1)) is helpful for finding the equation of a line when you know the slope and a point on the line.

By understanding these concepts, you can effectively find the equations of lines that are parallel or perpendicular to a given line, which is a valuable skill in various mathematical applications.