SOLUTION OF SIMPLE EQUATIONS.

152. The Solution of an Equation is the process of finding the value or values of the unknown quantity, or the roots of the equation.

153. A root of an equation is said to be verified, if the two members of the equation prove to be equal after the root has been substituted for the unknown quantity.

154. A simple equation may be solved, by transforming it in such a manner that the unknown quantity shall stand alone, and constitute one member of the equation; the other member will then be the value of the unknown quantity, or the root of the equation.

Let it be required to find the value of x in the equation

To verify this value of x, substitute it for x in equation (1);

Reducing each term to its simplest form, we obtain

the value of x is therefore verified.

155. It should here be observed that an equation of the first degree, containing but one unknown quantity, cannot have more than one root. For, whatever the equation may be, suppose it to

Find the value of x in each of the following equations:

33. a(x-b)+b(x−c)+c(x−a)=0. Ans. x=

α

Ans. x =

26

ab+ac+bc

a+b+c

PRODUCING EQUATIONS WHICH CONTAIN ONE UNKNOWN QUANTITY.

157. A Problem, in Algebra, is a question requiring the values of one or more unknown quantities from given conditions

158. The Solution of a problem is the process of finding the values of the unknown quantities.

159. Every problem in Algebra contains a statement of the relations between certain known and certain unknown quantities. When these relations are such as to furnish one or more equalities, the process of solution consists in expressing these equalities algebraically, and in solving the equations thus obtained.

160. There are two classes of problems which may be solved by the use of a single equation.

1st. Questions referring to a single unknown quantity.

2d. Questions referring to two or more unknown quantities, so related that when one is known, the others may be determined directly by the given conditions.

The following are examples of the first class:

1. What number is that the sum of whose third and fourth arts is 21?

Let x represent the number; then by the conditions,

2. A and B have each the same annual income. A's yearly expenses are $800 and B's $1000, and A saves as much in 5 years as B saves in 7 years; how much is the annual income?

Now by the conditions of the problem, we have

5(x800) 7 (x — 1000);

=

The following are examples of the second class:

3. Three men form a copartnership with a joint capital of $7200. A put in a certain sum, B put in three times as much as A, and C put in as much as both A and B ; how much did each man furnish?