Question

For the following exercises, use the descriptions of each pair of lines given below to find the slopes of Line 1 and Line 2. Is each pair of lines parallel, perpendicular, or neither?

14. Line 1: Passes through
(
0
,
6
)
and
(
3
,
−
24
)

Line 2: Passes through
(
−
1
,
19
)
and
(
8
,
−
71
)
15. Line 1: Passes through
(
−
8
,
−
55
)
and
(
10
,
89
)

Line 2: Passes through
(
9
,
−
44
)
and
(
4
,
−
14
)
16. Line 1: Passes through
(
2
,
3
)
and
(
4
,
−
1
)

Line 2: Passes through
(
6
,
3
)
and
(
8
,
5
)
17. Line 1: Passes through
(
1
,
7
)
and
(
5
,
5
)

Line 2: Passes through
(
−
1
,
−
3
)
and
(
1
,
1
)
18. Line 1: Passes through
(
0
,
5
)
and
(
3
,
3
)

Line 2: Passes through
(
1
,
−
5
)
and
(
3
,
−
2
)
19. Write an equation for a line parallel to
g
(
x
)
=
3
x
−
1
and passing through the point
(
4
,
9
)
.

20. Write an equation for a line parallel to
f
(
x
)
=
−
5
x
−
3
and passing through the point
(
2
,
-
12
)
.

21. Write an equation for a line perpendicular to
p
(
t
)
=
3
t
+
4
and passing through the point
(
3
,
1
)
.

22. Write an equation for a line perpendicular to
h
(
t
)
=
−
2
t
+
4
and passing through the point
(
-
4
,
-
1
)
.

Answer 1

Lines 14 and 15 are parallel, Lines 16 and 17 are neither parallel nor perpendicular, and Lines 18 are neither parallel nor perpendicular. Equations for parallel and perpendicular lines are provided for the respective scenarios.

Analysis of Pairs of Lines:

14. Line 1: (0,6) and (3,−24)

Line 2: (−1,19) and (8,−71)

Slope of Line 1:
`m_(1)`
`(-24-6)/(3-0) =-10`

Slope of Line 2:
`m_(2)`
`(-71-19)/(8-(-1)) =-10`

Conclusion: The slopes are equal (
`(m_(1) =m_(2) =-10)`, so the lines are parallel.

15. Line 1: (-8,-55) and (10,89)

Line 2: (9,-44) and (4,-14)

Slope of Line:1
`m_(1)`
`\frac {89-(-55)}{10-(-8)} =6`

Slope of Line:2
`m_( 2)`
`((-14)-(44))/(4-9) =6`

Conclusion: The slopes are equal
`m_(1) =m_(2) =6` ,so the lines are parallel

16. Line 1: (2,3) and (4,-1)

Line 2: (6,3) and (8,-5)

Slope of Line: 1
`m_(1) =(-1-3)/(4-2)=-2`

Slope of Line :2
`m_(2)= (5-3)/(8-6) =1`

Conclusion: The slopes are different
`(m_(1) \\eq m_(2)`
`)`,so the lines are neither parallel nor perpendicular.

17. Line 1: (1,7) and (5,5)

Line 2: (-1,-3) and (1,1)

Slope of Line 1:
`=m_(1)= (5-7)/(1-5) =(-1)/(2)`

Slope of Line 2:
`=m_(2)=(1-(-3))/(1-(-1)) =2`

Conclusion: The slopes are different
`(m_(1) \\eq m_(2))`,so the lines are neither parallel nor perpendicular.

18. Line 1: (0,5) and (3,3)

Line 2: (1,-5) and (3,-2)

Slope of Line 1:
`m_(1)= (3-5)/(3-0)=-(2)/(3) \\`

Slope of Line 2:
`m_(2)= ((-2)-(-5))/(3-1)=(3)/(2)`

Conclusion: The slopes are different
`(m_(1) \\eq m_(2))`,so the lines are neither parallel nor perpendicular.

Equations for Parallel and Perpendicular Lines:

1. Equation for line parallel to
`g(x) =3x-1` and passing through (4,90):

parallel lines have the same slopes, so the equation is
`y=3x+b`. plug in (4,9) to find b, yielding
`y=3x+5.`

2. Equation for line parallel to
`f(x)=-5x-3` and passing through (2,-12):

similar to the previous case,
`y =-5x+b`. substituting (2,-12) gives
`y=-5x-2.`

3 . Equation for line perpendicular to
`p(t)=3t+4`and passing through (3,-1):

Perpendicular lines have negative reciprocal slopes. thus
`y=-(1)/(3)t+b`, and using (3,1) gives .
`y=-(1)/(3)t+2`.

4. Equation for line perpendicular to
`h(t)=-2t+4`and passing through (-4,-1):

Following the pattern,
`y=-(1)/(2)t+b`, and with (-4,-1), the equation is
`y=-(1)/(2)t+1`.

The question probable may be:

For the following exercises, use the descriptions of each pair of lines given below to find the slopes of Line 1 and Line 2. Is each pair of lines parallel, perpendicular, or neither? 14. Line 1: Passes through ( 0 , 6 ) and ( 3 , − 24 ) Line 2: Passes through ( − 1 , 19 ) and ( 8 , − 71 ) 15. Line 1: Passes through ( − 8 , − 55 ) and ( 10 , 89 ) Line 2: Passes through ( 9 , − 44 ) and ( 4 , − 14 ) 16. Line 1: Passes through ( 2 , 3 ) and ( 4 , − 1 ) Line 2: Passes through ( 6 , 3 ) and ( 8 , 5 ) 17. Line 1: Passes through ( 1 , 7 ) and ( 5 , 5 ) Line 2: Passes through ( − 1 , − 3 ) and ( 1 , 1 ) 18. Line 1: Passes through ( 0 , 5 ) and ( 3 , 3 ) Line 2: Passes through ( 1 , − 5 ) and ( 3 , − 2 ) 19. Write an equation for a line parallel to g ( x ) = 3 x − 1 and passing through the point ( 4 , 9 ) . 20. Write an equation for a line parallel to f ( x ) = − 5 x − 3 and passing through the point ( 2 , - 12 ) . 21. Write an equation for a line perpendicular to p ( t ) = 3 t + 4 and passing through the point ( 3 , 1 ) . 22. Write an equation for a line perpendicular to h ( t ) = − 2 t + 4 and passing through the point ( - 4 , - 1 ) .