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1.Coincident Lines
2x + 5y + 24 = 0
4x + 10y + 12 = 0
Condition.
a1/a2 = b1/b2 = c1/c2  2/4=5/10=24/12

1/2=1/2=1/2

Hence,
they are all equal
Therefore, derived.

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Refer this Table

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An example of a set of inconsistent equations is x+2=4 and x+2=6.They don't have any common solution. 

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Intersecting, Coincident, Parallel

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Examples of consistent linear equations. 

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Examples of consistent linear equations. 

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Q) The sum of distance from A to B and B to C is 2 km and the sum of distance from A to B and B to C is 2000 km.
A) Its the same scenario with different values.

Let A to B = x

B to C  = y
Hence, 2x + 2y = 2000
y = 2000 - x / 2
= 1000 - x
The solutions are :
X  0         500  1000
Y  1000  500   0

x = 2000 - y / 2

= 1000 - y
The solutions are :
X  0         500  1000
Y  1000  500   0
Hence, the solutions are same as the scenario was same and the condition was different. This is an example of coincident lines.
Coincident lines have infinite solutions and hence consistent.

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Q: x – y = 8, 3x – 3y = 16
A: Given, x – y = 8 and 3x – 3y = 16
Standard Form of it: x-y-8=0 3x-3y-16=o
(a1/a2) = 1/3

(b1/b2) = -1/-3 = 1/3

(c1/c2) = -8/-16 = 1/2

Since, (a1/a2) = (b1/b2) ≠ (c1/c2)

The equations are parallel to each other and have no solutions. Hence, the pair of linear equations is inconsistent.

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Linear equations in two variables are equations which can be expressed as ax + by + c = 0, where a, b and c are real numbers and both a, and b are not zero.

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the two linear equation are in the same two variable x and y is called linear equation in two variable

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a1x + b1y + c1 = 0
a2x + b2y + c2 = 0
where a1, a2, b1, b2, c1, c2 are Real Numbers.