§ 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, Intermediate Algebra, 5e – Slide #2 Section 2.5

Point-Slope Form 145 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, Intermediate Algebra, 5e – Slide #3 Section 2.5

Point-Slope Form 145 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, Intermediate Algebra, 5e – Slide #4 Section 2.5

Point-Slope Form 145 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, Intermediate Algebra, 5e – Slide #5 Section 2.5

Equations of Lines 146 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, Intermediate Algebra, 5e – Slide #6 Section 2.5

Deciding which form to use: 146 y = mx + b y – y1 = m(x – x1) 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, Intermediate Algebra, 5e – Slide #7 Section 2.5

Modeling Life Expectancy 147 EXAMPLE 3 Go over example 3 See Figures 2.35(a) and 2.35(b) Use the data points to write the slope-intercept form of the equation of this line. Use the linear function to predict the life expectancy of an American man born in 2020. Find the slope of 0.215. The slope indicates that for each subsequent birth year, a man’s life expectancy is increasing by 0.215 years. Use the point-slope form to write the equation (model). y – 70.0 = 0.215(x – 20) Change to slope-intercept form y = 0.215x + 65.7 Life expectancy of men born in 2020 f(60) = 0.215(60) + 65.7 = 78.6 years Blitzer, Intermediate Algebra, 5e – Slide #8 Section 2.5

Modeling Life Expectancy 148 Check Points Do Check Point 3 on page 148 See Figure 2.36 Use the data points to write the slope-intercept form of the equation of this line. (round to 2 decimal places). Use the linear function to predict the life expectancy of an American woman born in 2020. Blitzer, Intermediate Algebra, 5e – Slide #9 Section 2.5

Modeling Life Expectancy 148 Check Points, continued Do Check Point 3 on page 148 See Figure 2.36 Use the data points to write the slope-intercept form of the equation of this line. (round to 2 decimal places). Use the linear function to predict the life expectancy of an American woman born in 2020. Blitzer, Intermediate Algebra, 5e – Slide #10 Section 2.5

Modeling Public Tuition Public Tuition: In 2005, the average cost of tuition and fees at public four-year colleges was $6130, and in 2010 it was $7610. Note that the known value for 2008 is $6530. Solution: The line passes through (2005, 6.1) and (2010, 7.6). Find the slope. Thus, the slope of the line is 296; tuition and fees on average increased by $296/yr. Substitute 5 for 2005, 10 for 2010, and 8 for 2008. Figure not in book

Modeling Public Tuition Modeling public tuition: Write the slope-intercept form of the of the line shown in the graph. What is the y-intercept and does it have meaning in this situation. This is the equation of the line in point-slope form. Modeling public tuition: Substitute 5 for 2005, 10 for 2010, and 8 for 2008. This is the equation of the line in slope-intercept form. This is the equation of the line in point-slope form. This is the equation of the line in slope-intercept form.

Modeling Public Tuition Using the slope-intercept form of the equation of the line shown in the graph. Use the equation to predict the average cost of tuition and fees at public four-year colleges in 2008. Substitute 2008 or 8 for x and compute y. Use the equation to predict the average cost of tuition and fees at public four-year colleges in 2015. Substitute 2015 or15 for x and compute y. The model predicts that the tuition in 2008 will be $7018 The model predicts that the tuition in 2015 will be $9090.

Modeling the Graying of America Write the slope-intercept form of the equation of the line shown in the graph. Use the equation to predict the median age of the U.S. population in 2020. Solution: The line passes through (10, 30.0) and (30, 35.3). Find the slope. The slope indicates that each year the median age of the U.S. population is increasing by 0.265 year. (30, 35.3) (10, 30.0)

Modeling the Graying of America Write the slope-intercept form of the equation of the line shown in the graph. Use the equation to predict the median age of the U.S. population in 2020. The slope indicates that each year the median age of the U.S. population is increasing by 0.265 year. This is the equation of the line in point-slope form. (30, 35.3) This is the equation of the line in slope-intercept form. (10, 30.0) A linear equation that models the median age of the U.S. population, y, x years after 1970.

Modeling the Graying of America Write the slope-intercept form of the equation of the line shown in the graph. Use the equation to predict the median age of the U.S. population in 2020. The slope indicates that each year the median age of the U.S. population is increasing by 0.265 year. A linear equation that models the median age of the U.S. population, y, x years after 1970. Use the equation to predict the median age in 2020. Because 2020 is 50 years after 1970, substitute 50 for x and compute y. (30, 35.3) (10, 30.0) The model predicts that the median age of the U.S. population in 2020 will be 40.6.

Slope and Parallel Lines Parallel and Perpendicular Lines Slope and Parallel Lines 1) If two nonvertical lines are parallel, then they have the same slope. 2) If two distinct nonvertical lines have the same slope, then they are parallel. 3) Two distinct vertical lines, both with undefined slopes, are parallel. y = 2x + 6 and y = 2x – 4 are parallel. y = -4x +5 and y = -4x + 3 are parallel. Blitzer, Intermediate Algebra, 5e – Slide #17 Section 2.5

Slope and Perpendicular Lines Parallel and Perpendicular Lines Slope and Perpendicular Lines 1) If two nonvertical lines are perpendicular, then the product of their slopes is -1. 2) If the product of the slopes of two lines is -1, then the lines are perpendicular. 3) A horizontal line having zero slope is perpendicular to a vertical line having undefined slope. y = 2x + 6 and y = -(1/2)x – 4 are perpendicular. y = -4x +5 and y = (1/4)x + 3 are perpendicular. Blitzer, Intermediate Algebra, 5e – Slide #18 Section 2.5

Parallel and Perpendicular Lines 148 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 – (-4) = -3(x –2) Substitute the given values y + 4 = -3(x –2) Simplify y + 4 = -3x + 6 Distribute y = -3x + 2 Subtract 4 from both sides Blitzer, Intermediate Algebra, 5e – Slide #19 Section 2.5

Parallel and Perpendicular Lines 149-150 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, Intermediate Algebra, 5e – Slide #20 Section 2.5

Parallel and Perpendicular Lines CONTINUED m = and the point (2,-4), Substitute the given values Simplify Distribute Subtract 4 from both sides Common Denominators Common Denominators Simplify Blitzer, Intermediate Algebra, 5e – Slide #21 Section 2.5

Parallel and Perpendicular Lines 149 -150 Do Check Point 4 on page 149 Find the equation that passes through (-2, 5) and is parallel to the line y = 3x+1 Blitzer, Intermediate Algebra, 5e – Slide #22 Section 2.5

Parallel and Perpendicular Lines 149 -150 Do Check Point 5 on page 150 Find the slope and equation of a line that passes through (-2, -6) and is perpendicular to x+3y=12 Standard format must be changed Blitzer, Intermediate Algebra, 5e – Slide #23 Section 2.5

DONE

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, Intermediate Algebra, 5e – Slide #25 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, Intermediate Algebra, 5e – Slide #26 Section 2.5

Example Modeling female officers In 1995, there were 690 female officers in the Marine Corps, and by 2010 this number had increased to about 1110. Refer to graph in Figure 3.48 on page 214. The slope of the line passing through (1995, 690) and (2010,1110) is The number of female officers increased, on average by about 28 officers per year. Estimate how many female officers there were in 2006. (2010, 1110) (1995, 690) // Write the slope-intercept form of the of the line shown in the graph.