3-7 Equations of Lines in the Coordinate Plane

Quick Review

Slope-intercept form is y = mx + b , y equals mx . plus b , comma  where m is the slope and b is the y-intercept.

Point-slope form is y − y 1 = m ( x − x 1 ) , y minus , y sub 1 . equals m , open . x minus , x sub 1 , close . comma  where m is the slope and ( x 1 , y 1 ) open , x sub 1 , comma . y sub 1 , close  is a point on the line.

Example

What is an equation of the line with slope − 5 negative 5  and y-intercept 6?

Use slope-intercept form: y = − 5 x + 6 . y equals , minus . 5 x plus 6 .

Example

What is an equation of the line through ( − 2 , 8 ) open negative 2 comma 8 close  with slope 3?

Use point-slope form: y − 8 = 3 ( x + 2 ) . y minus 8 equals . 3 open x plus 2 close .

Exercises

Find the slope of the line passing through the points.

34. ( 6 , − 2 ) , ( 1 , 3 ) open 6 comma negative 2 close comma open 1 comma 3 close
35. ( − 7 , 2 ) , ( − 7 , − 5 ) open negative 7 comma 2 close comma open negative 7 comma negative 5 close
36. Name the slope and y-intercept of y = 2 x − 1 . y equals 2 x , minus 1 .  Then graph the line.
37. Name the slope of and a point on y − 3 = − 2 ( x + 5 ) . y minus , 3 equals , minus . 2 open x plus 5 close .  Then graph the line.

Write an equation of the line.

38. slope − 1 2 , negative , 1 half , comma  y-intercept 12
39. slope 3, passes through ( 1 , − 9 ) open 1 comma negative 9 close
40. passes through (4, 2) and ( 3 , − 2 ) open 3 comma negative 2 close

3-8 Slopes of Parallel and Perpendicular Lines

Quick Review

Parallel lines have the same slopes.

The product of the slopes of two perpendicular lines is − 1 . negative 1 .

Example

What is an equation of the line perpendicular to y = 2 x − 5 y equals , 2 x minus 5  that contains ( 1 , − 3 ) ? open 1 comma negative 3 close question mark

• Step 1 Identify the slope of y = 2 x − 5 . y equals 2 x , minus 5 .

  The slope of the given line is 2.

• Step 2 Find the slope of a line perpendicular to y = 2 x − 5 . y equals 2 x , minus 5 .

  The slope is − 1 2 , negative , 1 half , comma  because 2 ( − 1 2 ) = − 1 . 2 . open , negative , 1 half , close . equals negative 1 .

• Step 3 Use point-slope form to write y + 3 = − 1 2 ( x − 1 ) . y plus , 3 equals , minus , 1 half , open x minus 1 close .

Exercises

Determine whether A B ↔ modified eh b with left right arrow above  and C D ↔ modified c d with left right arrow above  are parallel, perpendicular, or neither.

41. A ( − 1 , − 4 ) , B ( 2 , 11 ) , C ( 1 , 1 ) , D ( 4 , 10 ) eh open negative 1 comma negative 4 close comma b open 2 comma 11 close comma c open 1 comma 1 close comma d open 4 comma 10 close
42. A ( 2 , 8 ) , B ( − 1 , − 2 ) , C ( 3 , 7 ) , D ( 0 , − 3 ) eh open 2 comma 8 close comma b open negative 1 comma negative 2 close comma c open 3 comma 7 close comma d . open 0 comma negative 3 close
43. A ( − 3 , 3 ) , B ( 0 , 2 ) , C ( 1 , 3 ) , D ( − 2 , − 6 ) eh . open negative 3 comma 3 close . comma b open 0 comma 2 close comma c open 1 comma 3 close comma d open negative 2 comma negative 6 close
44. A ( − 1 , 3 ) , B ( 4 , 8 ) , C ( − 6 , 0 ) , D ( 2 , 8 ) eh . open negative 1 comma 3 close . comma b open 4 comma 8 close comma c . open negative 6 comma 0 close . comma d open 2 comma 8 close
45. Write an equation of the line parallel to y = 8 x − 1 y equals 8 x , minus 1  that contains ( − 6 , 2 ) . open negative 6 comma 2 close .
46. Write an equation of the line perpendicular to y = 1 6 x + 4 y equals , 1 sixth . x plus 4  that contains ( 3 , − 3 ) . open 3 comma negative 3 close .

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