eolas/zk/Solving_equations.md

---
tags: [algebra]
---

## Use inversion of operators

When solving equations we frequently make use of the
[ operator inversion rules](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 $9$: $$ 4(9) = 36$$