Two lines are parallel when they have the same gradient m and they do not intersect:

In this case, the system of equations formed by the two lines has no solutions.

If the lines have the same gradient and they intersect, then they must be the same line.

Two lines are perpendicular if they form a right angle with respect to each other. In this case, the rise of one line becomes the run of the other, with a sign change:

Suppose we have the straight lines y=3x−2 and y=2−3x. Where do the lines intersect?

Lines intersect when they have a point in common. That is, for some x:

Rearranging gives

If two lines do not intersect, then they must be parallel, since the definition of parallel is two straight lines that never meet.

If two lines are the same (possibly in different forms), then their intersection will all real numbers.

Suppose we have straight lines with equation 3y+2x−2=0 and 2y−3x+1=0. Where do the lines intersect?

We have the system of equations

There are two ways of solving this.

By substitution

Rearranging

Substituting this into 3y−3x+1=0:

So x=53​, which implies y=−32​⋅53​+32​=154​. So the intersection is (53​,154​).

By elimination

We can eliminate y from the equations by subtracting the second from the first:

So x=53​⇒y=154​ and the intersection is again (53​,154​).

We can use either of these methods to systems of equations with 2 equations and 2 unknowns.