Identify the parallel lines and perpendicular lines from the given set. Application
2x + y = 1
9x + 3y = 6
y = 3x
y = -3x
2y = 4x +6
Y = - x/2

The correct answer is: ∴ line e (2y = 4x +6) and line f (Y = - x/2) are perpendicular lines & line b (9x + 3y = 6) and line d (y = -3x) are parallel lines.

Step-by-step solution:-
We will simplify the given equations and compare the same with standard form of a straight line to find the value of m.
a. 2x + y = 1
∴ y = -2x + 1
Comparing the above equation with standard form of a line i.e. y = mx + c, we get- m = -2 ......................... (Equation i)
b. 9x + 3y = 6
∴ 3y = -9x + 6
∴ y = -3x + 2 ............................ (Dividing both sides by 3)
Comparing the above equation with standard form of a line i.e. y = mx + c, we get- m = -3 ......................... (Equation ii)
c. y = 3x
∴ y = 3x + 0
Comparing the above equation with standard form of a line i.e. y = mx + c, we get- m = 3 ......................... (Equation iii)
We know that slopes of perpendicular lines are negative reciprocals of each other.
and we observe that-
Slope of line e (2y = 4x + 6) = 2 ...................................................... (From Equation v)
∴ Slope of line e (2y = 4x + 6) = -1/ -1/2 …........................................ (Multiplying and dividing by -1/2)
∴ Slope of line e (2y = 4x + 6) = -1/ Slope of line f (y = -x/2) ........... (From Equation vi)
∴ Slope of line e and f are negative reciprocals of each other
∴ line e (2y = 4x +6) and line f (Y = - x/2) are perpendicular lines.
Also, We know that slopes of parallel lines are equal.
and we observe that-
Slope of line b (9x + 3y = 6) = Slope of line d (y = -3x) = -3 .............. (From Equation ii & iv)
∴ line b (9x + 3y = 6) and line d (y = -3x) are parallel lines.

d. y = -3x
∴ y = -3x + 0
Comparing the above equation with standard form of a line i.e. y = mx + c, we get- m = -3 ......................... (Equation iv)
e. 2y = 4x +6
∴ y = 2x + 3 .................................... (Dividing both sides by 2)
Comparing the above equation with standard form of a line i.e. y = mx + c, we get- m = 2 ......................... (Equation v)