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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

2.5 Assignment p. 150 (2-44 even, 50-56 even)