Use the Point-Slope Form to Write the Equation of a Line

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1 Math "Writing Equations of Lines" Objectives: * Use the point-slope form to write the equation of a line. * Use slope-intercept form to write the equation of a line. * Use slope as an aid when graphing. * General form of the equation of a line. Preliminaries: Slope of a Line: The slope of a line passing through points (x 1, y 1 ) and (x 2, y 2 ) is m = rise run = y 2 y 1 x 2 x 1 (where x 1 x 2 ) Also, we will use two facts from section 2.3: - Different lines having the same slope are parallel m 1 = m 2. - If the slopes of two lines are negative reciprocals, the lines are perpendicular m 1 = 1 m 2 or m 1 m 2 = 1. Definition: Standard (General) Form of a Linear Equation A linear equation in two variables is an equation that can be written in the form where A, B, and C are real numbers and A and B are not both zero. Ax + By = C Use the Point-Slope Form to Write the Equation of a Line Point-Slope Form: The equation of the line passing through (x 1, y 1 ) and with slope m is Example 1: (Using the point-slope form) Find an equation of the line that has slope 5 and passes trough (2, 6). Write the equation in standard form. Example 2: (Using the point-slope form) Find an equation of the line passing through ( 2, 5) and (, 3). Page: 1

2 Page: 2 Use Slope-Intercept Form to Write the Equation of a Line Slope-Intercept Form The equation of the line with slope m and y intercept (0, b) is Example 3: (Using the slope-intercept form) Use the slope-intercept form to write an equation of the line with slope 5 ( and y intercept 0, 1 ). Write the answer in standard form. Example : (Using the slope-intercept form) Find the slope and y-intercept of the line graphed below. line. Write the answer in standard form. y 5 Then use the slope-intercept form to write the equation of the x Example 5: (Using the slope-intercept form) Use the slope-intercept form to write an equation of the line that has the given slope and passes through the given point. a) slope 3; and passes through ( 2, 5) b) slope 2; and passes through ( 2, 8)

3 Page: 3 Example 6: (Applications) After purchasing a new drill press, a machine shop owner had his accountant prepare a depreciation worksheet for tax purposes. Depreciation Worksheet Drill press... $1,970 (new) Salvage value... $270 (in 10 years) a) Assuming straight-line depreciation, write an equation that gives the value (v) of the drill press after x years of use. b) Find the value of the drill press after years of use. Use Slope as an Aid When Graphing If we know the slope and the y intercept of a line, we can graph the line without constructing a table of solutions. Example 7: (Use slope as an aid when graphing) Find the slope and y-intercept of the line with the equation 3x 2y = and graph the line. y x

4 Page: Recognize Parallel and Perpendicular Lines Example 8: (Using slopes to recognize parallel and perpendicular lines) Show that the lines represented by 2y = 3x + 12 and 6x + y = 7 are parallel. Example 9: (Using slopes to recognize parallel and perpendicular lines) Show that the lines represented by 3x + 2y = 6 and 2x 3y = 6 are perpendicular. Example 10: (Using slopes to find the equation of parallel and perpendicular lines) Find the equation of the line that passes through ( 6, 3) and is perpendicular to the line y + 3x = 12. Write the equation in slope-intercept form.

5 Page: 5 Example 11: (Using slopes to find the equation of parallel and perpendicular lines) Find the equation of the line that passes through ( 2, 5) and is parallel to the line y = 8x 3. Write the equation in slope-intercept form. When we are asked to write the equation of a line, determine what we know about the graph of the line: its slope, its y-intercept, points it passes through, and so on. Then substitute the appropriate numbers into one of the following forms of a linear equation. Forms of a Linear Equation Standard Form A 0 and B 0 Slope-Intercept Form slope = m and y intercept is (0, b) Point-Slope Form slope = m and the line passes through (x 1, y 1 ) A Horizontal Line m = 0 and the y intercept is (0, b) A Vertical Line m = undefined and the x intercept is (a, 0)