Equations of the null and the infinite

Sometimes an equation will resolve to eliminate all its variables, this creates two results

1.Consider

5x + 25 = 5(x-50)

5x + 25 = 5x - 50

we subtract 5x from both sides and get

25=-50 or 0=75

In this case we know the equation doesn't make sense, since 0 is NOT equal to 75. When this happens we say the equation resolves to the empty set and that there is no solution.

2. Consider

5x + 25 = 5(x+5)

5x + 25 = 5x+ 25

we subtract 5x from both sides again

25=25 or 1=1

This solution is called the identity because the left side is exactly equal to the right side. You can substitute any number for x and arrive at the same answer, thus there are an infinite number of solutions. We can say that this equation resolves to the set of all reals, often notated as R.