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

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

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

$2x-5y+11=0$

$:. 5y=2x+11$

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

$:.$ 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

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

$3x-2y+15=0$

$:. 2y=3x+15$

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

$:.$ Slope

$m_1=3/2$

2. Slope of line

$2x+3y-4=0$

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

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

$:.$ 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

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

$x-y=1$

$:. y=x-1$

$:.$ Slope

$m_1=1$

2. Slope of line

$2x-3y+1=0$

$:. 3y=2x+1$

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

$:.$ 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

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

$x-2y+15=0$

$:. 2y=x+15$

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

$:.$ Slope

$m_1=1/2$

2. Slope of line

$3x+y-4=0$

$:. y=-3x+4$

$:.$ 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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4. Determine if two lines are perpendicular 5x+2y-11=0 and 2x-5y+11=0