# DETERMINE IF THE LINES ARE PARALLEL PERPENDICULAR OR NEITHER

How to decide whether the lines are parallel, perpendicular or neither ?

Parallel lines :

Parallel lines will have same slope and different y-intercepts.

m1 = m2

m1 and m2 are the slope of those two lines.

Perpendicular lines :

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

m1.m2 = -1

Coincident lines :

Coincident lines will have same slope and same y-intercepts.

The lines which has no relationship in between the slopes and y-intercepts can be considered neither.

Tell whether the lines are parallel, perpendicular, or neither.

Problem 1 :

Line 1 : through (-3, -7) and (1, 9)

Line 2 : through (-1, -4) and (0, -2)

Solution :

 Slope of line 1 :x1 = -3, y1 = -7 x2 = 1, y2 = 9m1 = (y2 – y1)/(x2 – x1)= (9 + 7)/(1 + 3)= 16/4= 4 Slope of line 2 :x1 = -1, y1 = -4x2 = 0,  y2 = -2m = (y2 – y1)/(x2 – x1)= (-2 + 4)/(0 + 1)= 2/1 = 2

They are not parallel, they are not perpendicular. So, the lines are neither.

Problem 2 :

Line 1 : through (2, 7) and (-1, -2)

Line 2 : through (3, -6) and (-6, -3)

Solution :

 Slope of line 1 :x1 = 2, y1 = 7 x2 = -1, y2 = -2m1 = (y2 – y1)/(x2 – x1)= (-2 - 7)/(-1 - 2)= -9/(-3)= 3 Slope of line 2 :x1 = 3, y1 = -6x2 = -6,  y2 = -3m2 = (y2 – y1)/(x2 – x1)= (-3 + 6)/(-6 - 3)= 3/-9 = -1/3

m1 x m2 = 3 (-1/3)

m1 x m2 = -1

So, the lines are perpendicular.

Determine whether the graphs of each pair of equations are parallel, perpendicular, or neither.

Problem 3 :

y = 3x + 4

y = 3x + 7

Solution :

y = 3x + 4 ----- (1)

y = 3x + 7 ------(2)

Comparing the above equations with slope intercept form

y = mx + b, we get

m1 = 3, m2 = 3

b1 = 4, b2 = 7

Both lines are having same slope.  but different y – intercepts. So, the given lines are parallel. Problem 4 :

y = -4x + 1

4y = x + 3

Solution :

y = -4x + 1 ------(1)

4y = x + 3

y = x/4 + 3/4   ------(2)

m = -4, m = 1/4

m1  m2 = -4 (1/4)

m1  m2 = -1

Since the product of the slopes is equal to -1. The lines are perpendicular. Problem 5 :

y = 2x - 5

y = 5x - 5

Solution :

y = 2x – 5   -----(1)

y = 5x - 5  -----(2)

Comparing (1) and (2)

y = mx + b

m = 2, m = 5

b = -5, b = -5

There is no relationship between slopes, so they are neither.

Problem 6 :

y = -1/3x + 2

y = 3x - 5

Solution :

y = -1/3x + 2  -----(1)

y = 3x - 5 -----(2)

y = mx + b

m1 = -1/3, m2 = 3

b = 2, b = -5

m1 = -1/3, m2 = 3

m⋅  m2 = (-1/3) ⋅ 3

Product of the slope is equal to -1. So, the lines are perpendicular. Problem 7 :

y = 3/5x - 3

5y = 3x - 10

Solution :

y = 3/5x – 3 ------(1)

5y = 3x - 10

y = 3/5x - 10 ------(2)

y = mx + b

m1 = 3/5, m2 = 3/5

b = -3, b = -10

Both lines have the same slope but have different y-intercepts. So, the lines are parallel.

Problem 8 :

y = 4

4y = 6

Solution :

y = 4 ------(1)

4y = 6

y = 6/4

y = 3/2------(2)

Both are horizontal lines and they will not intersect.

## Recent Articles 1. ### Solving Equations with Rational Coefficients Worksheet

Jun 05, 23 09:20 PM

Solving Equations with Rational Coefficients Worksheet

2. ### GL Assessment 11 Plus Maths Practice Papers

Jun 05, 23 08:05 AM

GL Assessment 11 Plus Maths Practice Papers