, A = bh (i.e., the formula for computing the area of a
rectangle), and Even though we use the same = sign in all cases, there are at least five different ways in which equations arise in mathematics, and it is important to understand the similarities and differences between each case.
This is the most basic way in which we use the = sign. It indicates that the quantities on either side are the "same thing"; that is, they have the same value. For example, when we say:
(4/2 + 3) - 6 = (3 - 5)/(4 - 2)
we mean that both sides compute to the same result (namely, -1).
If we say, "let t = 5" or "let f(x) = 3x - 2", we are defining a variable to take a given value, or a function to be given by the given formula (for all values of x). Although there is often good reason why we might make such a definition, such definitions do not require any explanation or justification.
Notice that this is not quite the same as the previous case, since it is the equation itself which tells us how to obtain the value for t or f(x). However, in the context of this definition, we can assume that both sides of t = 5 are then equal in value, or that f(x) = 3x - 2, for any chosen value of x. For example, this means that we use the either equation to substitute into another expression.
In Algebra, it is most common to write equations that may be true for some choices of the variables. For example, when we write the equation:
3x - 2 = -5
we do mean that the two sides are equal in value, but only for some, as yet unspecified, value of the variable (which in this case would be x = -1). Such equations do not usually arise as definitions, but are two expressions that happen to be equal in the context of the problem.
On the other hand, some other equations happen to be true for all values of the variables. We say that they express the fact that two expressions are "identically equal", and we say that the equation is an identity, for short. They express universal truths about:
, A = bh (i.e., the formula for computing the area of a
rectangle), and Such formulas are used to quickly and conveniently state mathematical theorems, that is, true statements about the nature of things. Once a theorem is proven to be true, it may be used to solve any given problem, much as you would use a definition, in that, it is not open to debate. However, before it is proven, it is quite different from a definition, since it requires a logical argument to demonstrate its validity.
A final way in which we may use an = sign is in a hypothetical situation. For example, one might pose the following equation
2x = -1
as an Algebra exercise. However, there is no such x value (any power is always positive), so we cannot really say that the two sides are equal in value. Many mathematical questions ask us to look for values of the variables for which the given equations would (in theory) be equal in value, but there is often no guarantee that such a solutions exist.
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