Suppose we have two lines:

A

_{1}x + B

_{1}y = C1

And A

_{2}x + B

_{2}y = C2

For these lines to be parallel the relation that is to be satisfied is:

A

_{1}/A

_{2}= B

_{1}/B

_{2}≠C

_{1}/C

_{2}

For these lines to be intersecting the relation to be satisfied is:

A

_{1}/A

_{2}≠B

_{1}/B

_{2}

Lastly, for these lines to be coinciding the condition is:

A

_{1}/A

_{2}= B

_{1}/B

_{2}=C

_{1}/C

_{2}

When two lines intersect they give a unique solution and such kind of System of Equations is known as consistent. Lines being perpendicular is a special case of Intersecting Lines where their slopes will result into a product of -1.

When two lines are coinciding, they are said to be having same Slope and x or y- intercept. These lines form a consistent system of equations and are also dependent. We get an infinite number of solutions for such equations.

When two lines are parallel they are said to have same slope but have different x or y- intercepts. In case of Parallel Lines we get no solutions and so this kind of system of equations is known as inconsistent system of equations.