## Interactive Algebra Review

(Based on the algebra reviews in Calculus Applied to the Real World , and Finite Mathematics and Calculus Applied to the Real World and )

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## 7. Equations

 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. Example x + y = 7 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 2 + 5 = 7. 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: 3x + 4 = 0 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.

 Solutions of Equations 1. Unique Solution This means that the equation has one, and only one, solution. Example The equation 3x + 12 = 0 has the unique solution x =   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 An equation can often have more than one solution. Example The equation x2 - 3x + 2 = 0 has the following two solutions x =   and x =     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.)

 x - 3 = 1 has the unique solution x = 3x + 1 = x has the unique solution x = x + 1 = 3x + 1 has the unique solution x = 1/x + 1 = 3 has the unique solution x = ax = 1 (a 0) has the unique solution x = ax + b = c (a 0) has the unique solution x =

By factoring the left-hand side (or any other method) find all possible solutions for x in the equations in the next group of exercises. When there are two solutions, enter each solution in a different box. When there is only one solution, enter the same solution in both boxes. Again, try to work mentally as far as possible. When you do decide to resort to pen and paper, be as meticulous as possible about the way you write mathematics, using as many steps as you like. This is the key to communicating your train of thought to the people assessing your work (they like that). Equally important, it helps you organize your thoughts more clearly.

 x2 - x = 0 has solution(s) , 2x2 = 2 has solution(s) , x2 - 5x + 6 = 0 has solution(s) , x2 + 2x + 1 = 0 has solution(s) ,

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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Last Updated: April, 1999