Exploring Advanced Line Equations: Parallel and Perpendicular Lines

Math A100 IUSB Sec. 3.6 detailed notes – Week 6 More Writing equations of lines Recall that in Section 3.5 that there are several different combinations of things they can give to you and ask you to write the equations of the line: 1) The line on a grid 2) Slope and the y-intercept 3) The slope and a point which is NOT the y-intercept (i.e. does NOT have “ b ” in it) 4) Two points, where one IS the y-intercept 5) Two points, in which neither one is the y-intercept 6) Special cases: Horizontal lines & Vertical lines We have another situation to look at in Section 3.6 – what if we introduce a second line to the grid? When you have two lines on the same grid (which we will go into more in Chapter 4), you have 4 possibilities: 1) The lines do not intersect  they are parallel 2) The lines intersect forming a right angle (90 degrees)  they are perpendicular 3) The lines intersect 4) The lines are drawn on top of one another  these are called “ concurrent ” We tell which of these four situations we have by looking at the equations of those lines: Option 1 : The lines do not intersect  they are parallel Parallel lines both have the same slant, otherwise they would cross. Another word for a line ’ s “ slant ” is “ slope ” !!  parallel lines have the same slope Example 1: What is the slope of the line that is parallel to 2x + 3y = 12 Slope of the Original line: 2x + 3y = 12  3y = -2x + 12  y = -2/3x + 4 Slope of the original line = - 2/3 Slope of the parallel line = - 2/3 Option 2 : The lines intersect forming a right angle (90 degrees)  they are perpendicular Perpendicular lines also have a special relationship in two ways:

1) For two lines to be perpendicular, if one line has a positive slope, the other line must have a negative slope 2) For two lines to make a right angle, the slope are reciprocals (i.e. the flip of each other)  For two lines to be perpendicular, their slopes are “ negative reciprocals ” i.e. opposite sign and flipped Example 1: What is the slope of the line that is perpendicular to 2x + 3y = 12 Slope of the Original line: 2x + 3y = 12  3y = -2x + 12  y = -2/3x + 4 Slope of the original line = -2/3 Slope of the perpendicular line = 3/2 Option 3: The lines intersect (but not perpendicular) When two lines cross, they obviously have different slopes. Unlike perpendicular lines, there is no special relationship between the slopes of lines that intersect in any general way. Option 4: The lines are drawn on top of one another  these are called “ concurrent ” When two lines are drawn on top of one another, they have the same equation, i.e. the same slope AND the same y-intercept. So, given the equations of two lines, we can tell the relationship between the two lines. Example 1: Are the following lines parallel, perpendicular, or neither? Line 1: 2x – 4y = 20 Line 2: -1x + 2y = 12 Find the slope of each line and compare: Line 1: 2x – 4y = 20  - 4y = -2x + 20  y = -2/-4x + 20/-4  Y = 1/2x - 5  slope = 1/2 Line 2: -1x + 2y = 12  2y = x + 12  y = 1/2x + 6  slope = 1/2 The slopes are the same  the lines are parallel Example 2: Are the following lines parallel, perpendicular, or neither? Line 1: 2x – 4y = 20

Line 2: 2x + y = 12 Find the slope of each line and compare: Line 1: 2x – 4y = 20  - 4y = -2x + 20  y = -2/-4x + 20/-4  Y = 1/2x - 5  slope = 1/2 Line 2: 2x + y = 12  1y = -2x + 12  y = -2/1x + 6  slope = -2 The slopes are opposite and flipped  the lines are perpendicular Example 3: Are the following lines parallel, perpendicular, or neither? Line 1: 2x – 4y = 20 Line 2: 3x + 2y = 12 Find the slope of each line and compare: Line 1: 2x – 4y = 20  - 4y = -2x + 20  y = -2/-4x + 20/-4  Y = 1/2x - 5  slope = 1/2 Line 2: 3x + 2y = 12  2y = - 3x + 12  y = -3/2x + 6  slope = -3/2 The slopes are different, ½ & -3/2, but no special relationship  the lines are neither parallel nor perpendicular Example 4: Are the following lines parallel, perpendicular, or neither? Line 1: 4x – 2y = 20 Line 2: 2x + y = 12 Find the slope of each line and compare: Line 1: 4x – 2y = 20  - 2y = -4x + 20  y = -4/-2x + 20/-4  Y = 2x - 5  slope = 2 Line 2: 2x + y = 12  y = - 2x + 12  slope = - 2 The slopes are opposites but not flipped  the lines are neither parallel nor perpendicular Example 5: Are the following lines parallel, perpendicular, or neither? Line 1: 4x – 2y = 20

Line 2: x - 2y = 12 Find the slope of each line and compare: Line 1: 4x – 2y = 20  - 2y = -4x + 20  y = -4/-2x + 20/-4  Y = 2x - 5  slope = 2 Line 2: x - 2y = 12  - 2y = - 1x + 12  y = 1/2x – 6  slope = 1/2 The slopes are flipped but not opposite  the lines are neither parallel nor perpendicular Now, we can a 7 th combination of things for writing the equation of a line: a point and a line that is either parallel or perpendicular to the line going thru the point. These are done just like the previous ones: 1) find the slope of the new line; 2) find “ b ” , then 3) put the parts together. NOTE: the only difference between these new combinations is how you are finding the slope!! Example 1: Write the equation of the line that goes thru (6, 7) and is parallel to 2x + 3y = 12. 1) Find the slope of the new line: Parallel  slopes are the same, so find the slope of the parallel line 2x + 3y = 12  3y = - 2x + 12  y = -2/3x + 4  slope = - 2/3 Parallel slope = - 2/3 , new line ’ s slope also = - 2/3 2) Find “ b ” Using “ y = mx + b ” to find “ b ”  y = 7; x = 6; m = -2/3 y = mx + b becomes 7 = ( -2/3 )(6) + b  7 = -4 + b  b = 11 3) Putting the parts together: m = -2/3 and b = 11 y = mx + b  y = - 2/3 x + 11  final answer Example 2: Write the equation of the line that goes thru (6, 7) and is perpendicular to 2x + 3y = 12. 1) Find the slope of the new line: Perpendicular  slopes are opposite & flipped, so find the slope of the perpendicular line 2x + 3y = 12  3y = - 2x + 12  y = -2/3x + 4  slope = - 2/3 Original slope = - 2/3, new line ’ s perpendicular slope = 3/2

2) Find “ b ” Using “ y = mx + b ” to find “ b ”  y = 7; x = 6; m = 3/2 y = mx + b becomes 7 = ( 3/2 )(6) + b  7 = 9 + b  b = - 2 3) Putting the parts together: m = 3/2 and b = - 2 y = mx + b  y = 3/2 x + - 2  final answer