BOOK II

CHAPTER I

EQUATIONS-EQUIVALENT EQUATIONS—TRANSFORMATION OF AN EQUATION INTO AN EQUIVALENT EQUATION-THE SOLUTION OF AN EQUATION OF THE FIRST DE

GREE IN ONE UNKNOWN QUANTITY

154. In Chapter VI, Book I, some simple equations and problems involving equations of one unknown quantity have already been solved and some of the properties of such equations discussed. It is now proposed to take up a more complete study of equations in one unknown quantity and of their properties.

155. Identity.-Identity is an abbreviated term for an identical equation, 278. In an identity any numerical values whatever can be assigned to the letters which enter in the two members of the equality, and if the indicated operations are performed the two members have the same numerical value.

(a+b)2= a2+2ab+b2

(a+b+c)2 = a2 + b2 + c2 + 2ab+2ac+2bc.

Replace now a, b, c by any numbers whatever, and perform the calculations indicated in each of these equalities. Both members in each case will be equal numbers, and, for this reason, these equations are called identities.

156. The Equation. When an equality can not be verified, except by assigning to one letter or to several letters particular values, the equality is called an equation of condition, 79, or simply an equation. The letters to which it is necessary to assign the particular values in order to render both members of the equality equal are called the unknown quantities.

1. Consider, for example, the equality,

Assign any value whatever to x (say x=7; then 5x+4=39, different from 19); then 5x + 4 takes in general a value different from 19; hence this equality is an equation in one unknown quantity. When the equation is written in the form

5+4=19

it is supposed that only such a value is assigned to x for which 5x+4 is equal to 19. The number 3 satisfies this condition. It remains to be proved that no other number will satisfy the same condition. 2. Let a second equality be

x+40= 13x;

when any value whatever is assigned to x, the two members of the equality will have in general different values. Accordingly, this equality is an equation in one unknown quantity. Give the values 5 and 8; then both members of the equation have the same values, respectively 65 and 104. It will be proved later that these two values are the only values of x for which the equality holds.

3. Consider the equation,

3.-2y+4= 6y4x+7.

If to x and y are assigned any values whatever, the two members of the equality will in general have different values. This equality is an equation in two unknown quantities.

157. Root of an Equation. To solve an equation is to find all the values, which, substituted for the unknown quantity, satisfy the equation.

Example:

has the root 3, and no other, as will be shown later.

has the two roots 5 and 8, and these only.

158. The Degree of an Equation. When both members of an equation are rational and integral in each of the unknown quantities, and the sum of the exponents of the unknown quantities in every term is found, that sum which is the greatest is the degree of the equation.

Consider, for example, the following equations: (1) 3x

(2) 4x-7y+3=6y-5x-7,

(3) x2+21=10.c,

(4) 4x-5xy-94y-11x+3

(1) is an equation of the first degree in one unknown quantity, (2) is an equation of the first degree in two unknown quantities, (3) is an equation of the second degree in one unknown quantity, (4) is an equation of the second degree in two unknown quantities.

159. Equivalent Equations. Two equations which have the same roots are called equivalent equations.

The following two theorems relate to the transformation of an equation into another which is equivalent to it.

THEOREMS CONCERNING THE TRANSFORMATION OF AN EQUATION INTO AN EQUIVALENT EQUATION

160. THEOREM I.—If the same finite quantity is added to or subtracted from both members of an equation, the result is a new equation equivalent to the first.

Let A and B represent the two members of the equation in one or more unknown quantities:

Let C be any expression which may involve the unknown quantities, but which remains finite for any finite values assigned to these unknown quantities. By adding C to both members of equation (1), equation (2) results.

It is necessary to prove that equations (1) and (2) are equivalent; that is to say, that every solution of equation (1) is a solution of equation (2,) and, conversely, every solution of equation (2) is a solution of equation(1).

in A and B, then A and B will take equal values, (1).

For these same values of the unknown quantities, C will have some finite value, and, consequently, A+ C and B+C will have equal values (281, 1), and equation (2) is satisfied. solution of equation (1) is a solution of equation (2).

Therefore any

Conversely, let xa', y=b', z = c', . . . . . be a solution of

values.

But for the same values of x, y, z,

the expression

C takes a certain finite value; and therefore A and B will be equal if AC is equal to B+ C; since, if the same finite quantity C is subtracted from equals (81, 2), the remainders are equal. Therefore any solution of equation (2) is a solution of equation (1), and equations (1) and (2) are equivalent.

By adding-C to both members of equation (1), a new equation, (3), is formed:

which is equivalent to equation (1).

161. Application. This theorem makes it possible to transfer a term in one member of an equation to the other member.

RULE. In order to transfer a term from one

member of an equation

to the other, it suffices to omit this term in the member in which it is found and write it in the other member with its sign changed.

For example, consider the equation

7x-510+4x.

In order to suppress and find

5 in the first member, add 5 to both members

7x-5+5=10+4x+5.

[Th. I, 160] Hence

The terms 5 and 5, whose sum is 0, cancel.

7x= 10+4x+5.

In order to remove 4 x from the second member, add both members.

But

4x

7x-4x 10+4x-4x+5.

4 x = 0, and therefore

7x-4x =

10+ 5.

All the terms which involve unknown quantities have been written on one side, and those which involve known quantities only, on the other; simplifying,

4x

=

3 x 15.

Then equations 7 x 10+5 and 3 x = 15 are equivalent to the original equation 7-5 10 4x (Th. I, 160).

162. REMARK.-If the signs of all the terms of both members of an equation are changed an equation remains which is equivalent to the first (see §82, 2); for by theorem I, $160, all of the terms may be transferred from the first member to the second, and those of the second to the first by changing the signs of the terms. Thus, the equation,

163. THEOREM II.-If both members of an equation are multiplied or divided by the same quantity, which has a finite and determinate value different from zero, a new equation, equivalent to the first, is formed.

Let A and B be the two members of an equation and C a finite, determinate quantity different from zero. It is necessary to show that the equations,

are equivalent.

A B

AC = BC,

According to the theorem in the preceding section, these equations are respectively equivalent to the equations,

It is sufficient therefore to show that these last equations are equivalent. They are equivalent because any system of values of the unknown quantities which satisfies equation(1) reduces the expression

to zero.

A-B

Since C is a finite quantity, this same system of values substituted for the unknown quantities will reduce the product,

C(A-B),

Conversely, any

to zero and will therefore satisfy equation (2). system of values which, when substituted for the unknown quantities, satisfies equation (2) will reduce the product

C(A-B)

to zero; and since Cis a finite quantity different from zero, this can happen only when A-B0 (875); that is, the equation,

This same system of values, therefore, when substituted for the unknown quantities in equation (1), will satisfy it.

REMARK.-The preceding discussion practically assumes that the multiplier Chas a determinate value, and that this value is neither zero nor infinity. If the multiplier is an expression which involves the unknown quantities it can become zero or infinity for a system of values of the unknown quantities, and, consequently, the reasoning in the preceding section no longer holds.