Algebra: What allows us to do the same thing to both sides of an equation?

Question

I understand that the expressions on both sides of an equal sign are the same entity, and I know that when you modify one side, the other must be changed because it is referring to the same thing. What I do not understand is why making a new equation (adding or taking away from an expression) allows one to know what an unknown represents. What about equations lets us do this?

Answer 1

I like to think of an unknown as a closed box, and our job is to discover what number is being concealed in the box. As you stated, an equation represents two expressions that are equivalent in value to one another. If one of the expressions contains an unknown, then we want to figure out what that unknown is through mathematical manipulation and the properties of an equation. In other words, we want to discover what's in the box by using math to open it up.

For example if I have an equation x + 2 = 5, then usually I am not very much concerned with what x + 2 is equivalent to. I want to know only what "x" is equivalent to! I want to find out what's inside that unknown box marked as "x". So I can subtract 2 from both sides of the equation and I will end up with x = 3. Through a little mathematical manipulation, I now know the exact value of "x". I have essentially opened up the box and seen that the unknown is equal to 3.

Now this is a trivial example, however as you get into more advanced equations, you will learn more advanced techniques to discover "what's in the box" and find the unknown, and that is very powerful!

Answer 2

When putting "like terms" together, you're making it easier for yourself. Remember; what you do to one side must be done to the other. There's also times when you have to "factor". Example: x^2-x-6=(x+2)(x-3).

Answer 3

An equation is essentially a statement that expression on the right-hand side and the expression on the left-hand side are equal to each other.

If you add a quantity to both sides, it's essentially: equals added to equals (that is, to the sides that were equal to each other in the first place) are equal.

But you must not divide by zero - that operation isn't allowed in this context.

Also, if you multiply both sides by zero, you are getting rid of the information, therefore it isn't advisable to do that.