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Find whether the lines representing the following pair of linear equations intersect at a point, are parallel or coincident: $2 x − 3 y + 6 = 0; 4 x − 5 y + 2 = 0$

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Parallel lines

Intersecting lines

Coincident lines

None of the above

answer is B.

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Detailed Solution

The given equations are

2 x − 3 y + 6 = 0

and

4 x + 5 y + 2 = 0

.
Comparing the given equations with standard linear equations

a 1 x + b 1 y + c 1 = 0

and

a 2 x + b 2 y + c 2 = 0

we get the following values.

a 1 = 2, b 1 = − 3, c 1 = 6, a 2 = 4, b 2 = 5, c 2 = 2

Solving

a 1 a 2

for the given expressions,

⇒ a 1 a 2 = 24 ⇒ a 1 a 2 = 12

Solving

b 1 b 2

for the given expressions,

⇒ b 1 b 2 = − 3 5

Solve

c 1 c 2

for the given expressions,

⇒ c 1 c 2 = 62 ⇒ c 1 c 2 = 31

Since

a 1 a 2 ≠ b 1 b 2

.
Thus, the given lines will have a unique solution and the lines intersect at a point.
Therefore, the correct option is 2.

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Find whether the lines representing the following pair of linear equations intersect at a point, are parallel or coincident: 2x−3y+6=0;4x−5y+2=0

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Find whether the lines representing the following pair of linear equations intersect at a point, are parallel or coincident: $2 x − 3 y + 6 = 0; 4 x − 5 y + 2 = 0$