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What is an Equation?
An equation is the statement that two mathematical expressions are equal. In other words, it consists of two mathematical expressions separated by an equal sign.
The letters that occur in an equation signify numbers. Some stand for well-known numbers, such as , c (the speed of light: 3108m/sec) or e (the base of natural logarithms: 2.71828 . . .). Some stand for variables or unknowns. Variables are quantities (such as length, height, or number of items) that can have many possible values, while unknowns are quantities whose values you may be asked to determine. The distinction between variables and unknowns is fuzzy, and mathematicians often use these terms interchangeably.
A solution to an equation in one or more unknowns is an assignment of numerical values to each of the unknowns, so that when these values are substituted for the unknowns, the equation becomes a true statement about numbers.
can be thought of as a linear equation in two unknowns, x and y. A solution to this equation is x = 2, y = 5, or (2, 5), since substituting 2 for x and 5 for y yields the true statement
Other solutions are (0, 7), (0.5, 6.5), and (-2, 9).
We could also think of x + y = 7 as an equation in two variables, as the numbers x and y could stand for quantities that can vary. For example, x could stand for the number of days per week you attend math class and y for the number of days per week you don't attend math class. The equation x + y = 7 then amounts to the statement that there are a total of seven days in the week. If you knew the number x, you could find the remaining unknown, y.
Note It's interesting to notice that x and y do not vary randomly-again, if you know x then you know y. We can say that the value of y depends on the value of x. It's also common to say that y is a function of x.
An equation in one unknown has exactly one variable, and the symbol x is traditionally reserved for that purpose (like most traditions, it is not strictly followed)
Example Here are a few equations in one unknown:
x2 - 3x + 2 = 0
4x4 + 11x2 + 9 = 0
x5 - 10x + 5 = 0 x0.5 -2x2 = 4x
There are two methods of solving an equation: analytical and numerical. To solve an equation analytically means to obtain exact solutions using algebraic rules. To solve it numerically means to use a computer or a graphing calculator to obtain solutions. In this appendix, we shall concentrate on the analytical approach.
We should point out that almost anything can happen when you try to solve an equation. Here are the possibilities, illustrated by examples.
1. Unique Solution
Sometimes the solution is not so easy to find. Often, it cannot be found at all analytically. An example is x5-10x+10 = 0, whose unique real solution can only be found numerically.
2. Two or More Solutions
Just as in the case of a unique solution, multiple solutions may not be easy to find. An example is x5-10x+5 = 0, whose three real solutions can only be found numerically.
3. No Solutions
The equation 4x4 + 11x2 + 9 = 0 has no real solutions whatever. Think for a while about why this should be the case before clicking here.
Mentally solve the following equations for x. (That is, try to solve them by writing down as little as possible.)
Now go over the examples and try some of the exercises in Section 7 of the Algebra Review of Calculus Applied to the Real World, or Finite Mathematics and Calculus Applied to the Real World.
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