# What is division property of equality

Equality is a property of an equation. An equations says that two expressions are equal and the symbol ‘=’ appears between those two expressions. In such cases the equality is not affected by adding or subtracting the same number on both sides. The number can also be 0 and still the equality is not disturbed.
Similarly the equality of an equation is not affected by multiplying or dividing both sides, again, by the same number. But an important point in the cases of multiplication or division the number cannot be 0. So with the restriction of 0, the equality of an equation is maintained when divided by the same number on both sides. This is what is known as division property of equality or simply as division property.

So, the division property of equality definition is, any equation maintains its equality status when divided on both sides by the same number except 0.

Let us illustrate some cases as division property of equality examples.
1) 4 = 4. When you divide by 2 on both sides you get 2 = 2 and when divided by 4 on both sides the result is 1 = 1. The answers are true in both cases. Also, the equality is not affected even when you divide by same negative numbers. For example, 4/(-2) =  4/(-2)? -2 = -2.

2) Let us try an equation involving simple variables. Say 3x = 6. When divided by 3 on both sides you get the solution as (3x)/(3) = (6)/(3) or x = 2. Same way, (3x)/(-3) = (6)/(-3) or -x = -2, which is also true.

3) Now let us discuss an example where many students tend to commit a mistake by not properly applying the division property.

Let, x2 = 4x be an equation. Now I divide both sides by x and solve x = 4. But x2 = 4x can be rewritten as a quadratic equation form as, x2- 4x = 0. A quadratic equation has always two solutions whereas we found only one solution. How and why the other solution is missing? What is the fallacy here?

The myth is, the step of dividing both sides of the equation x2 = 4xby x. Because if x = 0, the equation x2 = 4x is true and hence the variable can also take the value of 0.

When such being the case, the division by x is not valid and hence one should not have gone ahead with that step. The correct method of solving is x2 = 4x?x2 - 4x = 0?  x(x – 4) = 0. Now as per the zero product property there are two solutions as x = 0 and x = 4.