The properties used to solve an equation are the properties of the relationship of equality, reflexivity, symmetry and transitivity and the properties of operations.
These properties are as true in arithmetic and algebra as they are in propositional language.
This can be summarized as follows: If the same operation is performed on both sides of an equality, then the equality is still true.
Properties
The main properties of an equality are the logical properties that make it an equivalence relationship:
- the reflexive property: for all values of x, we always have x = x;
- the symmetric property: if x = y, then y = x;
- the transitive property: if x = y and y = z, then x = z.
The properties of operations (specific to each operation) are the following:
- Additive properties:
- If x = y, then x + z = y + z, for all values of z;
- If x + u = x + v, then u = v;
- Therefore, for all values of z : if x = y, then x − z = y − z.
- These properties can be generalized to all the operations as follows:
- If x = y, then x z = y z, for all values of z not equal to the absorbing element of the operation,
if x u = x v, then u = v, for all values of x not equal to the absorbing element of the operation.
- If x = y, then x z = y z, for all values of z not equal to the absorbing element of the operation,
Therefore, these properties are valid for the operations of multiplication and division, since the preceding restriction eliminates the case of division by zero.
Example
The variable y in the equation \(4x + 2y = 6x + 10\), can be isolated by following these steps:
| \(4x + 2y = 6x + 10\) | \(⇔\) | \(4x + 2y − 6x = 6x + 10 − 6x\) | Additive inverse |
| \(⇔\) | \((4x − 6x) + 2y = (6x − 6x) + 10\) | commutative and associative properties of addition |
|
| \(⇔\) | \(− 2x + 2y = 10\) | Simplify | |
| \(⇔\) | \(−2x + 2y − 2x = 10 − 2x\) | Additive inverse | |
| \(⇔\) | \((− 2x − 2x) + 2y = 10 − 2x\) | Commutative and associative properties of addition | |
| \(⇔\) | \(2y = 10 − 2x\) | calculation | |
| \(⇔\) | \(\dfrac{2y}{2} = \dfrac{(10 − 2x)}{2}\) | Division Property of Equality | |
| \(⇔\) | \(y = \dfrac{10}{2} − \dfrac{2x}{2}\) | distributive property of division over addition | |
| \(⇔\) | \(y = 5 − x\) | calculation |