Solving Equations Involving Parallel and Perpendicular Lines Examples

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1 Solving Equations Involving Parallel and Perpendicular Lines Examples. The graphs of y = x, y = x, and y = x + are lines that have the same slope. They are parallel lines. Definition of Parallel Lines In a plane, lines with the same slope are parallel lines. Also, all vertical lines are parallel.. Example Find the slope of a line parallel to the line whose equation is y x =. Parallel lines have the same slope. Find the slope of the line whose equation is y x =. To do so, write the equation in slope-intercept form (). y x = y = x + y = x + The slope of any line parallel to the given line is.. Example Find the slope of a line parallel to the line whose equation is y x = Parallel lines have the same slope. Find the slope of the line whose equation is y x =. To do so, write the equation in slope-intercept form (). y = x The slope of any line parallel to the given line is.

2 . Example Find an equation of the line that passes through (, 6) and is parallel to the line whose equation is y = x +. The slope is (notice that y = x + is in slope-intercept form. Use (, 6) and the slope to find the y-intercept. 6 = ( )() + b 6 = 8 +b 0 = b Substitution Property 0 An equation of the line is y = x +. Thought Provoker What is the relationship between the x- and y-intercepts of parallel lines? If the intercepts are nonzero, the ratio of the x- and y-intercepts of a given line is equal to the ratio of the x- and y-intercepts of any line parallel to the given line. 6. Example Find an equation of the line that passes through (, ) and is parallel to y x = Rewrite y x = into slope-intercept form y = x + The slope of all parallel lines must be. Find b by substituting into slope-intercept form = ( ) + b b = 0 Therefore y = x + 0 must be the equation of a line passing through (, ) and parallel to y x =.

3 7. Example Find an equation of the line that passes through (, ) and is parallel to x + y = 6. Rewrite x + y = 6 into slope-intercept form y = The slope of all parallel lines must be. 6 x + Find b by substituting into slope-intercept form = ( ) + b b = Therefore y = x and parallel to x + y = must be the equation of a line passing through (, ) 8. The graphs of y = x + and y = x + 6 are lines that are perpendicular. Notice how their slopes are related: ( )( ) = The product of their slopes is 9. Definition of Perpendicular Lines In a plane, two nonvertical lines are perpendicular if and only if the product of their slopes is. Any vertical line is perpendicular to any horizontal line. 0. Here is a way to show that the slopes of any two nonvertical perpendicular lines in a plane have a product of. Consider a line that is neither vertical nor horizontal, with slope s r (see graph). Now consider rotating the line Notice that the slope r of the new line (see graph) is. The product of s r r the slopes of the two lines is ( )( ) =. s s

4 . Example Find the slope of a line perpendicular to the line whose equation is y x =. y x = y = x + Write equation in slope-intercept form. The slope of the given line is. (m) = m = Let m stand for the slope of the perpendicular line. The slope of any line perpendicular to the given line is. Example Find the slope of a line perpendicular to the line whose equation is x 7y = 6. x 7y = 6 6 y = x 7 7 Write equation in slope-intercept form. The slope is 7 (m) = 7 7 m = Let m stand for the slope of the perpendicular line. The slope of any line perpendicular to the given line is 7

5 . Example Find an equation of the line that passes through (, 6) and is perpendicular to the line whose equation is y = x +. The slope of the given line is. (m) = m = Use (, 6) and the slope 6 = ( )() + b = b Let m stand for the slope of the perpendicular line. to find the y-intercept. The y-intercept is. An equation of the line is y = x + This could be written x + y =. Example Find an equation of a line passing through (, ) and is perpendicular to x + y = 6. y = x + 6 The slope of the given line is. (m) = Let m stand for the slope of the perpendicular line. m = Use (, ) and the slope to find the y-intercept. = ()( ) + b The y-intercept is 0. 0 = b An equation of the line is y = x This could be written x y = 0

6 . Example Find an equation of a line that passes through (, ) and is perpendicular x + y =. y = x + The slope of the given line is (m) = m =. Let m stand for the slope of the perpendicular line. Use (, ) and the slope to find the y-intercept. = ( )() + b = b The y-intercept is An equation of the line is y = x. This could be written x y = 6. Thought Provoker Is it possible that two lines represented by 6x y = and x + y = 7 are perpendicular? Rewrite in slope-intercept form. 6x y = y = 6x + y = x + x + y = 7 y = x y = x + Yes. They are perpendicular. 6

7 Name: Date: Class: Solving Equations Involving Parallel and Perpendicular Lines Worksheet Find the slope of a line that is parallel and the slope of a line that is perpendicular to each line whose equation is given.. y = x +. y = x. x y = 8. y = x 8 6. x = y y x = 0 State whether the graphs of the following equations are parallel, perpendicular, or neither. 7. x + y = 8. x + y = x + y = 0 x y = 9. y = x 0. y + x = y = x y x =. x 8y = x 6y = 0. y + x = y + x =. x + y =. x + y = x + y = 7 x y = 7

8 Find an equation of the line that passes through each given point and is parallel to the line with the given equation.. (, ); y = x 6. (, ); y = x (, ); x + y = 8. (0, 0); x y = Find an equation of the line that passes through each given point and is perpendicular to the line with the given equation. 9. (, 0); y = x (, ); x + y = 7. (, 6); x + y =. (0, ); 6x y = Find the value of a for which the graph of the first equation is perpendicular to the graph of the second equation.. y = ax ; y = x. y = ax + ; y x = 7. y = a x 6; x + y = 6 6. y + ax = 8; y = x + 8

9 Name: Date: Class: Solving Equations Involving Parallel and Perpendicular Lines Worksheet Key Find the slope of a line that is parallel and the slope of a line that is perpendicular to each line whose equation is given.. y = x + Parallel slope = ; Perpendicular slope =. y = x y = x + Parallel slope = ; Perpendicular slope =. y = x 8 y = x Parallel slope = ; Perpendicular slope =. 6y x = 0 y = 6 x 6 Parallel slope = ; Perpendicular slope = 6. x y = y = x Parallel slope = ; Perpendicular slope = x = y y = x Parallel slope = ; Perpendicular slope = State whether the graphs of the following equations are parallel, perpendicular, or neither. y = x + Both have same slope. 7. x + y = y = x 0 Lines are parallel. x + y = 0 9

10 8. x + y = x y = y = x + y = x The product of the slopes is ()( ) = Lines are perpendicular. 9. y = x y = x Lines have same slope. y = x y = x Lines are parallel. 0. y + x = y x = y = x + y = x + The product of the slopes is ( )( ) = Lines are perpendicular.. x 8y = y = x 8 8 Neither x 6y = 0 y = x. y + x = y + x = y = x + y = x + Neither. x + y = x + y = 7 9 y = x y = x + Lines have same slope. Lines are parallel.. x + y = x y = y = x + 6 y = x The product of the slopes is ( )( ) = Lines are perpendicular. 0

11 Find an equation of the line that passes through each given point and is parallel to the line with the given equation.. (, ); y = x Use m = and (, ) to find b. = () + b b = 6 y = x 6 6. (, ); y = x + 6 Use m = and (, ) to find b. = () + b b = 0 y = x 7. (, ); x + y = x + y = y = x + Use m = and (, ); to find b. = ( ) + b 7 7 b = y = x + 8. (0, 0); x y = x y = y = x Use m = and (0, 0); to find b. 0 = (0) + b b = 0 y = x

12 Find an equation of the line that passes through each given point and is perpendicular to the line with the given equation. 9. (, 0); y = x + 7 m = For a perpendicular line use m = and (, 0) 0 = ( ) + b b = y = x + 0. (, ); x + y = 7 x + y = 7 y = m = 7 x + For a perpendicular line use m = and (, ) = () + b b = y = x +. (0, ); 6x y = 6x y = y = x m = For a perpendicular line use m = = b = (0) + b y = x and (0, )

13 . (, 6); x + y = x + y = y = x + m = For a perpendicular line use m = and (, 6) 6 = () + b b = y = x Find the value of a for which the graph of the first equation is perpendicular to the graph of the second equation.. y = ax ; y = x Rewrite y = x in slope-intercept form. y = x Compare slopes with y = ax. If a has a perpendicular slope then (a) = a =. y = ax + ; y x = 7 Rewrite y x = 7 in slope-intercept form. y = x + 7 Compare slopes with y = ax +. If a has a perpendicular slope then (a) = a =

14 . y = a x 6; x + y = 6 Rewrite x + y = 6 in slope-intercept form. a y = x + 6 Compare slopes with y = x 6. a If a has a perpendicular slope then ( ) = a = 6. y + ax = 8; y = x + Rewrite y + ax = 8 in slope-intercept form. a 8 y = x + Compare slopes with y = x +. a If a has a perpendicular slope then ( ) = a =

15 Student Name: Date: Solving Equations Involving Parallel and Perpendicular Lines Checklist. On questions thru 6, did the student find the slope of a parallel line? a. Yes (0 points) b. out of 6 ( points) c. out of 6 (0 points) d. out of 6 ( points) e. out of 6 (0 points) f. out of 6 ( points). On questions thru 6, did the student find the slope of a perpendicular line? a. Yes (0 points) b. out of 6 ( points) c. out of 6 (0 points) d. out of 6 ( points) e. out of 6 (0 points) f. out of 6 ( points). On questions 7 thru, did the student correctly determine the type of graphs? a. Yes (0 points) b. 7 out of 8 ( points) c. 6 out of 8 (0 points) d. out of 8 ( points) e. out of 8 (0 points) f. out of 8 ( points) g. out of 8 (0 points) h. out of 8 ( points). On questions thru 8, did the student find the equation of a parallel line correctly? a. Yes (0 points) b. out of ( points) c. out of (0 points) d. out of ( points). On questions 9 thru, did the student find the equation of a perpendicular line correctly? a. Yes (0 points) b. out of ( points) c. out of (0 points) d. out of ( points) 6. On questions thru 6, did the student solve for a correctly? a. Yes (0 points) b. out of ( points) c. out of (0 points) d. out of ( points) Total Number of Points

16 Name: Date: Class: NOTE: The sole purpose of this checklist is to aide the teacher in identifying students that need remediation. It is suggested that teacher s devise their own point range for determining grades. In addition, some students need remediation in specific areas. The following checklist provides a means for the teacher to assess which areas need addressing.. Does the student need remediation in content (finding the slope of a parallel and perpendicular line when given a linear equation) for questions thru 6? Yes No. Does the student need remediation in content (identifying parallel and perpendicular lines when given a pair of linear equations) for questions 7 thru? Yes No. Does the student need remediation in content (finding the equation of a parallel line when given a point and a linear equation) for questions thru 8? Yes No. Does the student need remediation in content (finding the equation of a perpendicular line when given a point and a linear equation) for questions 9 thru? Yes No. Does the student need remediation in content (finding the value of the constant term when given a perpendicular line) for questions thru 6? Yes No A B C D F 0 points and above 6 points and above points and above 96 points and above 9 points and below Sample Range of Points. 6

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