1. 2.5
The Point-Slope Form of the Equation of a Line

2. Point-Slope Form of the Equation of a Line
The point-slope equation of a nonvertical line with slope m that passes through the point is
Blitzer, Algebra for College Students, 6e – Slide #2 Section 2.5

3. Point-Slope Form
EXAMPLE
Write the point-slope form and then the slope-intercept form of the equation of the line with slope -3 that passes through the point (2,-4).
SOLUTION
Substitute the given values
Simplify
This is the equation of the line in point-slope form.
Distribute
Subtract 4 from both sides
This is the equation of the line in slope-intercept form.
Blitzer, Algebra for College Students, 6e – Slide #3 Section 2.5

4. Point-Slope Form
EXAMPLE
Write the point-slope form and then the slope-intercept form of the equation of the line that passes through the points (2,-4) and (-3,6).
SOLUTION
First I must find the slope of the line. That is done as follows:
Blitzer, Algebra for College Students, 6e – Slide #4 Section 2.5

5. Point-Slope Form
CONTINUED
Now I can find the two forms of the equation of the line. In find the point-slope form of the line, I can use either point provided. I’ll use (2,-4).
Substitute the given values
Simplify
This is the equation of the line in point-slope form.
Distribute
Subtract 4 from both sides
This is the equation of the line in slope-intercept form.
Blitzer, Algebra for College Students, 6e – Slide #5 Section 2.5

6. Equations of Lines Equations of Lines Standard Form Ax + By = C
Slope-Intercept Form
y = mx + b
Horizontal Line
y = b
Vertical Line
x = a
Point-slope Form
Blitzer, Algebra for College Students, 6e – Slide #6 Section 2.5

7. Deciding which form to use:
Begin with the slope-intercept form if you know:
Begin with the point-slope form if you know:
The slope of the line and the y-intercept or
Two points on the line, one of which is the y -intercept
The slope of the line and a point on the line other than the y-intercept
Two points on the line, neither of which is the y-intercept
Blitzer, Algebra for College Students, 6e – Slide #7 Section 2.5

8. Parallel and Perpendicular Lines
1) If two lines are parallel, then they have the same slope.
2) If two nonvertical lines are perpendicular, then the product of their slopes is -1.
3) A horizontal line having zero slope is perpendicular to a vertical line having undefined slope.
Blitzer, Algebra for College Students, 6e – Slide #8 Section 2.5

9. Parallel and Perpendicular Lines
One line is perpendicular to another line if its slope is the negative reciprocal of the slope of the other line.
The following lines are perpendicular:
y = 2x + 6 and y = -(1/2)x – 4 are perpendicular.
y = -4x +5 and y = (1/4)x + 3 are perpendicular.
Blitzer, Algebra for College Students, 6e – Slide #9 Section 2.5

10. Parallel and Perpendicular Lines
Two lines are parallel if they have the same slope.
The following lines are parallel:
y = 2x + 6 and y = 2x – 4 are parallel.
y = -4x +5 and y = -4x + 3 are parallel.
Blitzer, Algebra for College Students, 6e – Slide #10 Section 2.5

11. Parallel and Perpendicular Lines
EXAMPLE
Write an equation of the line passing through (2,-4) and parallel to the line whose equation is y = -3x + 5.
SOLUTION
Since the line I want to represent is parallel to the given line, they have the same slope. Therefore the slope of the new line is also m = -3. Therefore, the equation of the new line is:
y – 2 = -3(x – (-4))
Substitute the given values
y – 2 = -3(x + 4)
Simplify
y – 2 = -3x - 12
Distribute
y = -3x - 10
Add 2 to both sides
Blitzer, Algebra for College Students, 6e – Slide #11 Section 2.5

12. Parallel and Perpendicular Lines
Your Turn
Write an equation of the line passing through (-2,5) and parallel to the line whose equation is y = -3x + 1.
Blitzer, Algebra for College Students, 6e – Slide #12 Section 2.5

13. Parallel and Perpendicular Lines
EXAMPLE
Write an equation of the line passing through (2,-4) and perpendicular to the line whose equation is y = -3x + 5.
SOLUTION
The slope of the given equation is m = -3. Therefore, the slope of the new line is , since Therefore, the using the slope m = and the point (2,-4), the equation of the line is as follows:
Blitzer, Algebra for College Students, 6e – Slide #13 Section 2.5

14. Parallel and Perpendicular Lines
CONTINUED
Substitute the given values
Simplify
Distribute
Subtract 4 from both sides
Common Denominators
Common Denominators
Simplify
Blitzer, Algebra for College Students, 6e – Slide #14 Section 2.5

15. Parallel and Perpendicular Lines
Your Turn
Write an equation of the line passing through (3,-5) and perpendicular to the line whose equation is x + 4y =8.
Blitzer, Algebra for College Students, 6e – Slide #15 Section 2.5

16. 2.5 Assignment
p. 150 (2-44 even, even)