1. Introduction
The slopes of parallel lines are always equal, whereas the slopes of perpendicular lines are always opposite reciprocals. It is important to be able to determine whether lines are parallel or perpendicular, but the creation of parallel and perpendicular lines is also important. In this lesson, you will write the equations of lines that are parallel and perpendicular to a given line through a given point.
6.1.3: Working with Parallel and Perpendicular Lines

2. Key Concepts
You can write the equation of a line through a given point that is parallel to a given line if you know the equation of the given line. It is necessary to identify the slope of the given equation before trying to write the equation of the line that is parallel or perpendicular.
Writing the given equation in slope-intercept form allows you to quickly identify the slope, m, of the equation.
6.1.3: Working with Parallel and Perpendicular Lines

3. Writing Equations Parallel to a Given Line Through a Given Point
Key Concepts, continued
If the given equation is not in slope-intercept form, take a few moments to rewrite it.
Writing Equations Parallel to a Given Line Through a Given Point
Rewrite the given equation in slope-intercept form if necessary.
Identify the slope of the given line.
Write the general point-slope form of a linear equation: y – y1 = m(x – x1).
Substitute the slope of the given line for m in the general equation.
Substitute x and y from the given point into the general equation for x1 and y1.
Simplify the equation.
Rewrite the equation in slope-intercept form if necessary.
6.1.3: Working with Parallel and Perpendicular Lines

4. Key Concepts, continued
Writing the equation of a line perpendicular to a given line through a given point is similar to writing equations of parallel lines.
The slopes of perpendicular lines are opposite reciprocals.
6.1.3: Working with Parallel and Perpendicular Lines

5. Key Concepts, continued
Writing Equations Perpendicular to a Given Line Through a Given Point
Rewrite the given equation in slope-intercept form if necessary.
Identify the slope of the given line.
Find the opposite reciprocal of the slope of the given line.
Write the general point-slope form of a linear equation: y – y1 = m(x – x1).
Substitute the opposite reciprocal of the given line for m in the general equation.
Substitute x and y from the given point into the general equation for x1 and y1.
Simplify the equation.
Rewrite the equation in slope-intercept form if necessary.
6.1.3: Working with Parallel and Perpendicular Lines

6. Key Concepts, continued
The shortest distance between two points is a line.
The shortest distance between a given point and a given line is the line segment that is perpendicular to the given line through the given point.
6.1.3: Working with Parallel and Perpendicular Lines

7. Key Concepts, continued
Writing Equations Perpendicular to a Given Line Through a Given Point
Follow the steps outlined previously to find the equation of the line that is perpendicular to the given line through the given point.
Find the intersection between the two lines by setting the given equation and the equation of the perpendicular line equal to each other.
Solve for x.
Substitute the x-value into the equation of the given line to find the y-value.
Find the distance between the given point and the point of intersection of the given line and the perpendicular line using the distance formula,
6.1.3: Working with Parallel and Perpendicular Lines

8. Common Errors/Misconceptions
attempting to identify the slope of the given line without transforming the equation into slope-intercept form
incorrectly identifying the slope of the given line
incorrectly finding the slope of the line parallel to the given line
incorrectly identifying the slope of the line perpendicular to the given line
improperly substituting the x- and y-values into the general point-slope equation
6.1.3: Working with Parallel and Perpendicular Lines

9. Guided Practice Example 1
Write the slope-intercept form of an equation for the line that passes through the point (5, –2) and is parallel to the graph of 8x – 2y = 6.
6.1.3: Working with Parallel and Perpendicular Lines

10. Guided Practice: Example 1, continued
Rewrite the given equation in slope-intercept form.
8x – 2y = 6 Given equation
–2y = 6 – 8x Subtract 8x from both sides.
y = –3 + 4x Divide both sides by –2.
y = 4x – 3 Write the equation in slope intercept form.
6.1.3: Working with Parallel and Perpendicular Lines

11. Guided Practice: Example 1, continued
Identify the slope of the given line.
The slope of the line y = 4x – 3 is 4.
6.1.3: Working with Parallel and Perpendicular Lines

12. Guided Practice: Example 1, continued
Substitute the slope of the given line for m in the point-slope form of a linear equation.
y – y1 = m(x – x1) Point-slope form
y – y1 = 4(x – x1) Substitute m from the given equation.
6.1.3: Working with Parallel and Perpendicular Lines

13. Guided Practice: Example 1, continued
Substitute x and y from the given point into the equation for x1 and y1.
y – y1 = 4(x – x1) Equation
y – (–2) = 4(x – 5) Substitute (5, –2) for x1 and y1.
6.1.3: Working with Parallel and Perpendicular Lines

14. Guided Practice: Example 1, continued Simplify the equation.
y – (–2) = 4(x – 5) Equation with substituted values for x1 and y1
y – (–2) = 4x – 20 Distribute 4 over (x – 5).
y + 2 = 4x – 20 Simplify.
y = 4x – 22 Subtract 2 from both sides.
The equation of the line through the point (5, –2) that is parallel to the equation 8x – 2y = 6 is y = 4x – 22.
6.1.3: Working with Parallel and Perpendicular Lines

15. ✔ Guided Practice: Example 1, continued
This can be seen on the following graph. ✔
6.1.3: Working with Parallel and Perpendicular Lines

16. Guided Practice: Example 1, continued
6.1.3: Working with Parallel and Perpendicular Lines

17. Guided Practice Example 3
Find the point on the line y = 4x + 1 that is closest to the point (–2, 8).
6.1.3: Working with Parallel and Perpendicular Lines

18. Guided Practice: Example 3, continued
Find the line perpendicular to the given line, y = 4x + 1, that passes through the point (–2, 8).
6.1.3: Working with Parallel and Perpendicular Lines

19. Guided Practice: Example 3, continued
Identify the slope of the given line.
The slope of the line y = 4x + 1 is 4.
6.1.3: Working with Parallel and Perpendicular Lines

20. Guided Practice: Example 3, continued
Find the opposite reciprocal of the slope of the given line.
The opposite of 4 is –4.
The reciprocal of –4 is
6.1.3: Working with Parallel and Perpendicular Lines

21. Guided Practice: Example 3, continued
Substitute the opposite reciprocal for m in the point-slope form of a linear equation.
y – y1 = m(x – x1) Point-slope form
Substitute m from the given equation.
6.1.3: Working with Parallel and Perpendicular Lines

22. Guided Practice: Example 3, continued
Substitute x and y from the given point into the equation for x1 and y1.
Equation
Substitute (–2, 8) for x and y1.
6.1.3: Working with Parallel and Perpendicular Lines

23. Guided Practice: Example 3, continued Simplify the equation.
Equation with substituted values for x1 and y1
Distribute over (x – (–2)).
Add 8 to both sides.
6.1.3: Working with Parallel and Perpendicular Lines

24. Guided Practice: Example 3, continued
The equation of the line through the point (–2, 8) that is perpendicular to the graph of y = 4x + 1 is
This can be seen on the following graph.
6.1.3: Working with Parallel and Perpendicular Lines

25. Guided Practice: Example 3, continued
6.1.3: Working with Parallel and Perpendicular Lines

26. Guided Practice: Example 3, continued
Find the intersection between the two lines by setting the given equation equal to the equation of the perpendicular line, then solve for x.
Set both equations equal to each other.
Subtract 1 from both sides.
6.1.3: Working with Parallel and Perpendicular Lines

27. Guided Practice: Example 3, continued
Add to both sides.
Divide both sides by
6.1.3: Working with Parallel and Perpendicular Lines

28. Guided Practice: Example 3, continued
Substitute the value of x back into the given equation to find the value of y.
y = 4x Given equation
Substitute for x.
Simplify.
6.1.3: Working with Parallel and Perpendicular Lines

29. Guided Practice: Example 3, continued
The point on the line closest to (–2, 8) is the point
6.1.3: Working with Parallel and Perpendicular Lines

30. Guided Practice: Example 3, continued
Calculate the distance between the two points using the distance formula.
Distance formula
Substitute values for (x1, y1) and (x2, y2) using (–2, 8) and
6.1.3: Working with Parallel and Perpendicular Lines

31. Guided Practice: Example 3, continued Simplify. Evaluate squares.
6.1.3: Working with Parallel and Perpendicular Lines

32. ✔ Guided Practice: Example 3, continued Reduce to lowest terms.
The distance between the point of intersection and the given point is units, or approximately 3.6 units. ✔
6.1.3: Working with Parallel and Perpendicular Lines

33. Guided Practice: Example 3, continued
6.1.3: Working with Parallel and Perpendicular Lines