Two sets, P & Q, satisfy the following conditions P={(x,y):y=2x+5}, Q={(x,y):y=mx+c} & P ∩ Q= Empty set. White down the value of m..& a possible...

If P ∩ Q= Empty set, the sets don' have any elements in
common. That means that the system formed from the
equations:

y=2x+5

and

y
= mx + c

doesn't have any
solution.

The geometric explanation is that the lines whose
equations are y=2x+5 and y = mx + c do not have any intercepting
point.

We'll re-write the
equations:

2x - y = -5 (1)

mx
- y = -c (2)

We'll multiply (1) by -m and (2) by
2:

-2mx + my = 5m (3)

2mx - 2y
= -2c (4)

We'll add (3) +
(4):

-2mx + my + 2mx - 2y = 5m -
2c

We'll eliminate like
terms:

my - 2y = 5m - 2c

We'll
factorize by y to the left side:

y(m-2) = 5m -
2c

We'll divide by m-2:

y =
(5m - 2c)/(m-2)

It is obvious that the ratio doesn't exist
if the denominator is cancelling. So, y has no value and also
x!

For the denominator m-2 to be zero, we'll form the
equation:

m - 2 = 0

We'll add
2:

m = 2

So, P ∩
Q= Empty set, if and only if m = 2.