---
categories:
  - Mathematics 
tags: [algebra]
---
## Use inversion of operators

When solving equations we frequently make use of the [ operator inversion rules](../Prealgebra/Inversion%20of%20operators.md) to find the solutions.

### Example: inversion of addition

For example, the equation $9 = 3 + x$ has the solution $6$ ($x$ is equal to $6$). To arrive at this, we can use the inverse of the main operator in the equation (addition): $9-3 = 6$.

### Example: inversion of subtraction

Now consider $19 = x - 3$.  The solution to this equation is  $22$ ($x$ is equal to $22$). To arrive at this, we can use the inverse of the main operator in the equation (subtraction): $19 + 3 = 22$.

### Example: inversion of division

The equation we want to solve: 
$$\frac{x}{6} = 4$$

Now we invert it by multiplying the denominator by the quotient: $6\cdot 4 = 24$. Therefore:
$$ \frac{24}{6} = 4$$
The solution is $24$

### Example: inversion of multiplication

The equation we want to solve:
$$4x = 36$$ 
Now we invert it by dividing the product by the coefficient: 
!Add link to 'coefficient'

$$\frac{36}{4} = 9$$ 

Therefore the solution is $9$:
$$ 4(9) = 36$$
