Algebra is a branch of mathematics that uses mathematical statements to describe relationships between things that change over time. These variables include things like the relationship between supply of an object and its price. When we use a mathematical statement to describe a relationship, we often use letters to represent the quantity that varies, since it is not a fixed amount. These letters and symbols are referred to as variables. (See the Appendix One for a brief review of constants and variables.)
But first an example: An arithmetic problem might look like this: 7 + ? = 10 and we are to find a value for the ? that makes this equation true. From inspection we see that the ? represents 3 since 3 is the only number that makes the equation true. So, 7 + 3 = 10 is the answer and that means the ? = 3.
Another Example in algebraic form: A + B = C and we are given the facts that A=7 and C=10 so what is the value of B? We can re-write the equation substituting what is known and we have as our algebraic equation: 7 + B = 10 and this looks a lot like our arithmetic equation. We can use a simple algebraic rule to find the value of B which right now is the unknown to us. The rule says you can subtract the same amount from each side of the equation to help get the answer. The idea is to eliminate everything on one side of the equation except the unknown, B in our example. Our unknown is B so subtract 7 from each side of the equation to get B by itself on one side, like this.
(7 - 7) + B = 10 - 7, subtracting 7 from each side of the equation results in; 0 + B = 3 or simplified B = 3, the unknown we were looking for.
The Rules of Algebra:
The rule of "symmetry": This simply means each side of the equals sign = must be equal. For example 3 = 3, or A = B. Since a = b (whatever value A is, B is the same value). So, if A = 7 then we also know B = 7 since A = B. Since this is true then B = A is true as well.
The commutative rules: The order of terms does not matter. As we saw above, A = B and B = A. So, A + B = B + A. Either way organize the terms it is still true. Since this is true it is also true that; A * B = B* A. This is called the commutative rule of multiplication. A multiplied by B is equal to B multiplied by A.
The inverse of adding: The inverse of an operation like addition for example, undoes the operation. For example if we add 5 to something then immediately subtract 5 we have inversely added 5. So, 2 + 5 - 5 = 2 + 0 = 2. Right now that may make no sense to you and you might say what's the point. The point is that in algebra with an unknown it is often helpful to do inverse adding to eliminate terms of an equation.
Two rules for equations:
1. You can add the same value to each side of the equation. For example, given: A = B then it is also true that A + C = B + C. C can be any number and the equation is still true.
2. You can multiply each side of an equation with the same value and the equation will still be true. For example if A = B then it is also true that A*C = B*C.
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