HOW TO TELL IF LINES ARE PARALLEL OR PERPENDICULAR WITH EQUATIONS

• If two lines are parallel, then their slopes will be equal

m₁ = m₂

• If two lines are perpendicular, then product of their slopes will be equal to -1.

m₁ x m₂ = -1

Examine whether these lines are parallel or perpendicular !

Problem 1 :

2x - y + 7 = 0 and 4y - 2x + 3 = 0

Solution :

Given, 2x - y + 7 = 0 and 4y - 2x + 3 = 0

2x - y + 7 = 0

-y = -7 - 2x

y = -(-7 - 2x)

y = 2x + 7

Slope intercept equation of a line y = mx + b, we get

m = slope

m₁ = 2

4y - 2x + 3 = 0

4y = 2x - 3

m₂ = 1/2

The slopes are m₁ = 2 and m₂ =  1/2.

= 2 × 1/2

= 1

Since, the slope of the lines m₁ and m₂ are neither.

Problem 2 :

y = (-2/9)x - 23 and 2y - 9x + 3 = 0

Solution :

Given, y = (-2/9)x - 23 and 2y - 9x + 3 = 0

y = (-2/9)x - 23

Slope intercept equation of a line y = mx + b, we get

m = slope

m₁ = -2/9

2y - 9x + 3 = 0

2y = 9x - 3

m₂ = 9/2

The slopes are m₁ = -2/9 and m₂ =  9/2.

= -2/9 × 9/2

= -1

Since, the slope of the lines m₁ and m₂ are perpendicular.

Problem 3 :

2x + 5y + 17 = 0 and 5x = 2y - 13

Solution :

Given, 2x + 5y + 17 = 0 and 5x = 2y - 13

2x + 5y + 17 = 0 

5y = -2x - 17

Slope intercept equation of a line y = mx + b, we get

m = slope

m₁ = -2/5

5x = 2y - 13

2y = 5x + 13

m₂ = 5/2

The slopes are m₁ = -2/5 and m₂ =  5/2.

= -2/5 × 5/2

= -1

Since, the slope of the lines m₁ and m₂ are perpendicular.

Problem 4 :

(1/2)x + (1/3)y - 4 = 0 and 3y - 2x + 3 = 0 

Solution :

Given, (1/2)x + (1/3)y - 4 = 0

Slope intercept equation of a line y = mx + b, we get

m = slope

m₁ = -3/2

3y - 2x + 3 = 0 

3y = 2x - 3

m₂ = 2/3

The slopes are m₁ = -3/2 and m₂ =  2/3.

= -3/2 × 2/3

= -1

Since, the slope of the lines m₁ and m₂ are perpendicular.

Problem 5 :

3x - y + 7 = 0  and y - 2x - 43 = 0 

Solution :

Given, 3x - y + 7 = 0 and y - 2x - 43 = 0

3x - y + 7 = 0

-y = -7 - 3x

y = -(-7 - 3x)

y = 3x + 7

Slope intercept equation of a line y = mx + b, we get

m = slope

m₁ = 3

y - 2x - 43 = 0

y = 2x + 43

m₂ = 2

The slopes are m₁ = 3 and m₂ =  2.

Since, the slope of the lines m₁ and m₂ are neither.

Problem 6 :

x - 5y + 9 = 0  and 4y - x + 3 = 0 

Solution :

Given, x - 5y + 9 = 0  and 4y - x + 3 = 0 

x - 5y + 9 = 0

-5y = -x - 9

Slope intercept equation of a line y = mx + b, we get

m = slope

m₁ = -1/5

4y - x + 3 = 0 

4y = x - 3

m₂ = 1/4

The slopes are m₁ = -1/5 and m₂ =  1/4.

Since, the slope of the lines m₁ and m₂ are neither.