Question

Is each line parallel, perpendicular, or neither parallel nor perpendicular to the line −x+4y=20?

Drag each choice into the boxes to correctly complete the table.

Answer 1

-x+4y=8: Parallel
4x+y=-1: Perpendicular
y=-1/4x+6: neither
y=-4x-3: Perpendicular

Answer 2

we have

`-x+4y=20`

Isolate the variable y

`4y=x+20`

`y=(1)/(4)x+5`

the slope m of the given line is

`m=(1)/(4)`

we know that

If two lines are parallel , then their slopes are the same

so

`m1=m2`

if two lines are perpendicular, then the product of their slopes is equal to minus one

so

`m1*m2=-1`

we will proceed to verify each case to determine the solution

case A)
`-x+4y=8`

Isolate the variable y

`4y=x+8`

`y=(1)/(4)x+2`

the slope is

`m2=(1)/(4)`

Compare the slope of the line of the case A) with the slope of the given line

`m1=(1)/(4)` -----> slope given line

`m2=(1)/(4)` ----> slope line case A)

`m1=m2` --------> the lines are parallel

case B)
`4x+y=-1`

Isolate the variable y

`y=-4x-1`

the slope is

`m2=-4`

Compare the slope of the line of the case B) with the slope of the given line

`m1=(1)/(4)` -----> slope given line

`m2=-4` ----> slope line case B)

`m1*m2=(1)/(4)*-4=-1` --------> the lines are perpendicular

case C)
`y=-(1)/(4)x+6`

the slope is

`m2=-(1)/(4)`

Compare the slope of the line of the case C) with the slope of the given line

`m1=(1)/(4)` -----> slope given line

`m2=-(1)/(4)` ----> slope line case C)

`m1\\eq m2`

`m1*m2\\eq-1`

therefore

the line case C) and the given line are neither parallel nor perpendicular

case D)
`y=-4x-3`

the slope is

`m2=-4`

Compare the slope of the line of the case D) with the slope of the given line

`m1=(1)/(4)` -----> slope given line

`m2=-4` ----> slope line case D)

`m1*m2=(1)/(4)*-4=-1` --------> the lines are perpendicular

the answer in the attached figure