39. Shew that

(i)

(ii)

(iii)

a2

EXAMPLES.

b2

101

WILLIAM J. HUSSEY

LIBRARY OF

(a - b) (a − c) (1 + ax) * (b − c) (b − a) (1 + bx)

+

c2

1

(c − a) (c − b) (1 + cx) ̄ (1 + ax) (1 + bx) (1 + cx) °

α

+

b

(a - b) (a − c) (1 + ax) (b − c) (b − a) (1 + bx)

+

с

(c − a) (c − b) (1 + cx) ̄ ̄ (1 + ax) (1 + bx) (1 + cx) °

40. Simplify

1

− a) (c − b) (1 + cx) ̄ (1 + ax) (1 + bx) (1 + cx) *

+

(a − b) (a − c) (a — d) ̄ (b − c) (b − d) (b − a) ̄ (c–d) (c− a) (c — b)

(d − a) (d — b) (d −c)

= a+b+c+d.

is equal to zero if r be less than n-1, to 1 if r = n − 1, and to

CHAPTER IX.

EQUATIONS. ONE UNKNOWN QUANTITY.

114. A STATEMENT of the equality of two algebraical expressions is called an equation; and the two equal expressions are called the members, or sides, of the equation.

When the equality is true for all values of the letters involved the equation is, as we have already said, called an identity, the name equation being reserved for those cases in which the equality is only true for certain particular values of the letters involved.

For the sake of distinction, a quantity which is supposed to be known, but which is not expressed by any particular arithmetical number, is usually represented by one of the first letters of the alphabet, a, b, c, &c., and a quantity which is unknown, and which is to be found, is usually represented by one of the last letters of the alphabet x, y, z, &c.

115. We shall in the present chapter only consider equations which contain one unknown quantity.

To solve an equation is to find the value or values of the unknown quantity for which the equation is true; and these values of the unknown quantity are said to satisfy the equation, and are called the roots of the equation.

Two equations are said to be equivalent when they have the same roots.

An equation which contains only one unknown quantity, x suppose, and which is rational and integral in x, is said to be of the first degree when a occurs only in the first power; it is said to be of the second degree when a2 is the highest power of x which occurs; and so on.

Equations of the first, second and third degrees are however generally called simple, quadratic and cubic equations respectively.

116. In the solution of equations frequent use is made of the following principles.

I. An equation is equivalent to that formed by adding the same quantity to both its members.

For it is clear that A+m=B+m when, and only when, A = B.

II. Any term may be transposed from one side of an equation to the other, provided its sign be changed. Let the equation be

Add

then

that is,

a+b-c=p-q+r.

-p+q-r to both sides;
a+b-c-p+q−r=p−q+r−p + q−r,

a+b-c-p+q-r=0.

We thus have an equation equivalent to the given equation, but with the terms p, q,+r changed in sign and transposed.

By means of transposition all the terms of any equation may be written on one side of the sign of equality and zero on the other side.

III. An equation is equivalent to that formed by multiplying (or dividing) each of its members by the same quantity which is not equal to zero.

For, if A =B, it is clear that mA = mB. Conversely, if mA = mB, that is m (AB) = 0, it follows that A - B=0, since m is not zero. Hence mA = mB when, and only when, A = B.