# What is a reciprocal in math

## Reciprocal of an equation

Reading time: 5 min

We can use the reciprocal value not only for fractions, but also when transforming equations. To do this, we need to invert the left side of the equation as well as the right side of the equation.

If there is a fraction on each side, then we proceed as follows:

**Example equation:**

Both sides are equal in value, and that is

**Reciprocal of the equation:**

Both sides are still the same in value, with

If two numbers a and b are identical, then, of course, also because they are exactly the same numerical values. (Apart from that, the reciprocal cannot be calculated because it is not defined.)

Therefore, “taking the reciprocal” is a valid refor- mation of the equation as long as neither side of the equation is 0, that is, **it does not change the solution set of the equation**. The value for the unknown x is not changed.

For example, if we want to solve the simple equation:

Then we take the reciprocal on both sides and get:

### The reciprocal of the sum on one side of the equation

We can also form the reciprocal value if there is a sum on one side of the equation. Then the total sum must be taken into account for the reciprocal value. Example:

### Reciprocal value as multiple transformation of the equation

It's no magic that the reciprocal of an equation works. We can prove it as a multiple transformation of the equation:

We recognize that is equivalent (equal in value) to.

The reciprocal of an equation is nothing more than a multiple multiplication or division of the corresponding values.

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