A pair of simultaneous equations may have no solutions, one unique pair of solutions, or an infinite number of solutions.

**No solutions**

When two simultaneous equations contradict each other, there is no solution.

**Example**

3x + 2y = 10

6x + 4y = 22

When we double the first equation we get 6x + 4y = 20.

This contradicts the second equation which says that 6x + 4y = 22 not 20.

There can be no solution.

**One unique solution**

**Example**

3x + 2y = 7

6x + 5y = 16

When we solve this pair, we can double the first equation and then subtract this from the second equation.

This give us the solution x = 1, y = 2.

This is the only solution.

**An infinite number of solutions**

When two simultaneous equations are essentially the same as each other, there are an infinite number of solutions.

**Example**

3x + 2y = 10

6x + 4y = 20

When we double the first equation we get the second equation. So really we only have one equation, to which there are an infinite number of solutions.

If x = 0, y = 5.

If x = 2, y = 2.

If x = -5, y = 12.5.

Whatever value is chosen for x, you can always find a value for y that works.