The Concept of Equality
Audience: PreAlgebra, Beginning Algebra

The concept of equality is of utmost importance in Mathematics. But I see many students who have a less than sufficient understanding of equality. Once equality is afforded understanding and respect, a student becomes prepared to begin digesting mathematics.
There's more to equality than saying x = 12. We must learn to expand our understanding of equality so that we can know how to effectively manipulate mathematical "expressions" while preserving their "value" and effectively manipulate mathematical "equations" while preserving their "equality". That's right; Expressions have a value that, AT TIMES, must be preserved and Equations have an equality that ALWAYS must be preserved. And preservation is accomplished by an application of the understanding of the concept of equality.
In order to understand how this concept applies to both expressions and equations, we must understand the difference between an expression and an equation. Let's look at expressions and equations and in doing so we will see how equations are actually made up of expressions.
So what is an expression? Examples of expressions are: 4, 3x, 3x + 2, 2x + 3x, 4x/5, 1/3, 2x + 5 + 3x^{2} + y. Notice none of these expressions have an equals(=) sign  that's the chief difference between expressions and equations. But don't stop there  let's notice a few other things about these expressions; two of them have a value that doesn't change (they are said to be constant). One of those two is 4. Can you tell what the other one is? The rest of them have a value that depends on one or more variables. Four of them have a single term (monomial), two of them have two terms (binomial), and one of them has four terms (quadnomial). The word polynomial can be used to describe all of them. The word polynomial is a pronoun and is used to identify most expressions. It's like the word "car". A car can be a sedan, compact, truck, suv, etc. But we use the word car as a pronoun to refer to all of these. In the same way we use the word polynomial as a pronoun to refer to most expressions. An expression can have any number of variables and also any number of terms (terms are separated by a plus"+" or a minus"").
How about equations? All equations include an equals(=) sign. That's what makes them equations. Some examples are: 2 + 3 = 5, 2x + 6 = 10, 5x^{2} + x  10 = 0. At the risk of being redundant notice all of these equations have an equals(=) sign. Also notice the equations have expressions on both sides of the equals sign.
Let's be more detailed in the way we look at equations. Let's think of equations as having a left side and a right side. Each side contains an expression made up of one or more terms. That's ALWAYS true  each side of an equation contains an expression. The expression may be a constant, like 12, or it may depend on a variable, like 2x + 4x. 12 is an expression that can only represent one number, it's unchanging  it's always 12. 2x + 4x is an expression that can represent any number. All we need to do to obtain any number from 2x + 4x is adjust the value of x. If x = 6, then 2x + 4x = 36 (2*6 + 4*6). If x = 1, then 2x + 4x = 6. You get the idea I hope. If you don't get it, please reread the preceding before continuing. If you do get it, that's awesome!
Let's make an equation out of the last two expressions just discussed. That equation is 2x + 4x = 12.
With equations the goal is to get one of the variables by itself on one side of the equation. Once that is accomplished we've solved the equation for that variable. 2x + 4x = 12 is an equation with only one variable, so our goal is get x all by itself on one side of the equation.
Now let's see how the concept of equality applies and guides us as we go through the steps to solve the equation. First we'll work on the left side of the equation, here we'll change the configuration of the expression (simplify the expression), but we won't change its value  the concept of equality at work.
2x + 4x = 12 (simplify the left side of the equation)
6x = 12
These two equations have identically the same meaning. On the left side, where something changed, we preserved the value of 2x + 4x while rewriting it as 6x. That means that 2x + 4x is the same as 6x for any value of x. The value of 2x + 4x depends on x. We can't remove the dependency on x, but we can rewrite the expression to express it in simpler terms. We rewrote it as 6x. Both of these expressions, 2x + 4x and 6x, represent the exact same value no matter what the value of x is. By doing this we've simplified the left hand side of the equation and preserved the equality of the original equation. (See proof that 2x + 4x = 6x).
Now let's work on both sides of the equation at the same time. You can do any mathematical operation (plus "+", minus "", times "*", divide "/") to both sides of an equation. By doing the same operation with the same operator to both sides, we preserve the equality stated in the original equation. We could add 5 to both sides of 6x = 12. The resulting equation would be 6x + 5 = 17 (6x plus 5 is 6x + 5 and 12 plus 5 is 17). But doing this would not lead us closer to our goal. Our goal is to get x all by itself on one side of the equation. Adding 5 to both sides of the equation leads us away from our goal. What would lead us closer to our goal? Dividing (operation) both sides by 6 (operator) will put us at our goal.
6x = 12 (divide both sides of the equation by 6)
x = 2
Notice the values on both sides of the equation have changed, but changing these values by the same amount preserves the equality stated in the original equation. When you change (operate on) both sides of the equation by the same amount (do the same operation with the same operator, 6 in this case), you preserve the equality of the original equation. The resulting equation allows us to read the value of x that satisfies the original equation; x = 2. This means that if the value of x is 2, then the original equation, 2x + 4x = 12, is satisfied  it's a true statement.
In summary, we use the concept of equality to preserve the value of expressions when we simplify expressions. And we use the concept of equality to preserve the equality expressed in an equation when we do the same operation with the same operator to both sides of the equation.
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