Three Possibilities

When you've got two linear equations, there are three possible scenarios that can occur when you're asked to find their intersection.

Description	Identifying	Example
The lines intersect at exactly one point.	Your solution ends up with two values.	y = 3x + 1 y = 2x + 2 If you apply the substitution method from the previous section, you'll get (1, 4) as the solution.
The lines don't intersect, i.e. they're parallel.	At some point during your calculations, you reach an obviously false statement, e.g. 3 = -2.	y = 3x + 4 y = 2x - 1 If you apply the substitution method from the previous section, you'll get 4 = -1. That's not true for any value of x so your answer would be, "There is no solution."
The lines intersect at every point, i.e. they overlap.	At some point during your calculations, you reach a statement that's always true, e.g. -2 = -2.	y = 3x + 4 y = 3x + 4 If you apply the substitution method from the previous section, you'll get 4 = 4. That's true for every value of x so your answer would be, "All real numbers."

Description

Identifying

Example

The lines intersect at exactly one point.

Your solution ends up with two values.

y = 3x + 1
y = 2x + 2

If you apply the substitution method from the previous section, you'll get (1, 4) as the solution.

The lines don't intersect, i.e. they're parallel.

At some point during your calculations, you reach an obviously false statement, e.g. 3 = -2.

y = 3x + 4
y = 2x - 1

If you apply the substitution method from the previous section, you'll get 4 = -1. That's not true for any value of x so your answer would be, "There is no solution."

The lines intersect at every point, i.e. they overlap.

At some point during your calculations, you reach a statement that's always true, e.g. -2 = -2.

y = 3x + 4
y = 3x + 4

If you apply the substitution method from the previous section, you'll get 4 = 4. That's true for every value of x so your answer would be, "All real numbers."