1. Determine if two lines are perpendicular `5x+2y-11=0` and `2x-5y+11=0`

Solution:
When two lines are perpendicular, their slopes are opposite reciprocals of one another or the product of their slopes is -1.

1. Slope of line `5x+2y-11=0`

`5x+2y-11=0`

`:. 2y=-5x+11`

`:. y=-(5x)/(2)+11/2`

`:.` Slope `m_1=-5/2`


2. Slope of line `2x-5y+11=0`

`2x-5y+11=0`

`:. 5y=2x+11`

`:. y=(2x)/(5)+11/5`

`:.` Slope `m_2=2/5`

Now, `m_1*m_2=-5/2 xx 2/5=-1`

Here product is -1, so these two lines are perpendicular

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2. Determine if two lines are perpendicular `3x-2y+15=0` and `2x+3y-4=0`

Solution:
When two lines are perpendicular, their slopes are opposite reciprocals of one another or the product of their slopes is -1.

1. Slope of line `3x-2y+15=0`

`3x-2y+15=0`

`:. 2y=3x+15`

`:. y=(3x)/(2)+15/2`

`:.` Slope `m_1=3/2`


2. Slope of line `2x+3y-4=0`

`2x+3y-4=0`

`:. 3y=-2x+4`

`:. y=-(2x)/(3)+4/3`

`:.` Slope `m_2=-2/3`

Now, `m_1*m_2=3/2 xx -2/3=-1`

Here product is -1, so these two lines are perpendicular

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3. Determine if two lines are perpendicular `x-y=1` and `2x-3y+1=0`

Solution:
When two lines are perpendicular, their slopes are opposite reciprocals of one another or the product of their slopes is -1.

1. Slope of line `x-y=1`

`x-y=1`

`:. y=x-1`

`:.` Slope `m_1=1`


2. Slope of line `2x-3y+1=0`

`2x-3y+1=0`

`:. 3y=2x+1`

`:. y=(2x)/(3)+1/3`

`:.` Slope `m_2=2/3`

Now, `m_1*m_2=1 xx 2/3=2/3!=-1`

Here product is not -1, so these two lines are not perpendicular

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4. Determine if two lines are perpendicular `x-2y+15=0` and `3x+y-4=0`

Solution:
When two lines are perpendicular, their slopes are opposite reciprocals of one another or the product of their slopes is -1.

1. Slope of line `x-2y+15=0`

`x-2y+15=0`

`:. 2y=x+15`

`:. y=(x)/(2)+15/2`

`:.` Slope `m_1=1/2`


2. Slope of line `3x+y-4=0`

`3x+y-4=0`

`:. y=-3x+4`

`:.` Slope `m_2=-3`

Now, `m_1*m_2=1/2 xx -3=-3/2!=-1`

Here product is not -1, so these two lines are not perpendicular

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