# Mathematical Equations and Their Properties

In mathematics, when we see the = sign, we have an equation. In an equation,we mean that the expression on the left hand side equals the expression on the right hand side. The equation, $x + 2 = 5$ means that there is a value for $x$ that makes the equation true (or false). Mathematical equations can always (5x + 2x = 7x), sometimes (y + 3 = 5), or never (2z + z = 3z + 1) be true. Image via Annenberg Media

An equation is like a balance. If the objects on the left balance with the objects on the right side ( if a = b), then to maintain the balance, whatever you do on one side you also do to the other side.

• If you add a weight on the right hand side, then you also add the same weight to the other side. If a = b, then a + c = b + c.
• If you subtract  weight on one side, then you also subtract the same weight to the other side. If a = b, then ac = bc.
• If you multiply the weight on one side with a certain number, then you also have to multiply the same weight to the other side. If a = b, then ac = bc.
• If you divide the weight on one side with a certain number, then you also have to divide the same weight to the other side. If a = b, and c not equal to 0, then a/c = b/c.
• The weight of an object is equal to the weight of itself, so we conclude that a = a (reflexive property).
• If the weight of the first object is equal to the weight of the second object, then it is obvious that the weight of the second object is the same as that of the first object. This means if a = b, then b = a (symmetric property).
• If we have three objects,  the weight of the first object is the same as the weight of the second object, and the weight of the second object is the same as that of the third object, then the weight of the first object equals the weight of the third object. That is, if a = b, and b = c,  then a = c (transitive property).
The first two bullets are called the addition property of equality and the second two bullets are called multiplication property of equality. 