is unlimitedly great. Now if (for generality) we suppose (1) and (2) to be true, when a is greater than 1, it is clear that their first members will be impossible or imaginary (since 1x will be negative, and that V1 expresses its square 1. x root), at the same time that the second member of (1) is negative, and that of (2) positive; consequently, negative and positive expressions are sometimes signs of impossibility.

2. Because the squares of the first members of (1) and (2) when a is not greater than 1, it follows

are 1

x and

1

1

that (according to the preceding generalization) if x is greater than 1, we must find the square of 1 by rejecting the sign of the root, which is in conformity to what has (previously) been shown.

3. If we divide the members of (1) by 1-x (since

-X

1.3x2

etc., which is the same as (2); consequently (be

2.4 cause when a is greater than 1), the right member of (1) being negative and that of (2) positive, it results that the sign of the imaginary expression 1— is changed by dividing it by the negative number 1, which is in conformity to the rule of signs in Division.

4. Squaring (1) and (2), we have 1-x=

ly, if æ is greater than 1, the positive expressions (1 - 2

Xx

c.) and (1 (1 − + etc.) will be represented by the nega

SECTION XIII.

EQUATIONS.

(1.) AGREEABLY to Def. 6, p. 2, when the equality or equivalence of expressions is denoted by writing = (the sign of equality) between them, the result is called an equation; noticing that the expression which precedes the sign is called the first member, and that the remaining expression is called the second member; also, if the members are composed of monomials connected by either of the signs,, the monomials are called the terms of the equation. Thus, 22 ax + b = c − d is an equation, having aa2 ax + b and с d for its first and second members, x2, ax, b, c, and d being the terms of the equation.

(2.) Equations are (generally) composed of numbers or letters or both, which represent numbers or quantities; the first letters, a, b, c, etc., being used to stand for known numbers or quantities, while the last letters, x, y, z, etc., stand for unknown numbers or quantities. An equation which contains only one unknown letter is sometimes called a single equation.

(3.) If either member of an equation is the same as the other (under the same or a different form) or of a process indicated in the other, or if the terms of the equation are known, and given so as to make the members equal to each other, then the equation is called an identical equation. Thus, x2 a2

3x + 5 = 2x + x + 5, (x + a)2 = x2 + 2xa + a2, x

= x + a, 5 + 3 − 1 = 4+ 3, are identical equations.

(4.) If a single equation is such that it can not be satisfied, or its members made equal to each other, except by assigning one or more particular values to the unknown letter (contained in it), it is called a determinate equation; and any particular value of the unknown letter which satisfies the equation is called its root. The discovery of the roots is called the solution of the equation, and when the roots are

x

found the equation is said to be solved. Thus, 2x+35 is a determinate equation, because 1 is the only value of a or root which satisfies the equation; similarly, the equation a2. =6 has 3 and 2 for its roots, and because they are the only values of a which can satisfy the equation, it follows that 3 and 2 are the only roots that the equation can have.

(5.) If an equation contains more than one unknown letter, and becomes a determinate equation when particular values are given to all the unknown letters but one, then we shall call the (given) equation an indeterminate equation. Thus, the equation 35y= 7 is an indeterminate equation; for if 1, 2, 3, etc., are successively put for y, the equation will be reduced to the determinate equations 3x-5=7, 3x - 10 = 7, 3x - 15 = 7, etc.

=

(6.) If an equation contains one or more unknown letters, and is such that it can be reduced so as to consist of a finite number of monomial terms, into which the unknown letters enter only as factors with positive integral indices, it is called an algebraic equation; but if the equation can not be thus reduced, it is called a transcendental equation. Thus, 3x- 5 =2x+3, 3x2 + 2x 12, x3-2x2 + 9x 10, 3x 5y = 6, x25xy 3220, are algebraic equations; and 3 10, x = 13, a2 = b, z=d, are transcendental equations.

=

=

(7.) An algebraic equation which contains one or more unknown letters is said to be of the same degree as the greatest number which can be formed by adding the indices of the unknown letters which enter as factors into its separate terms, noticing that the equation is supposed to be reduced so as to consist of monomial terms, into which the unknown letters enter (only) as factors with positive integral indices. Thus, 3x720, 4x + 3y = 14 are of the first degree, 324x8, 2xy + 3y 120 are of the second degree, and so on.

(8.) An equation of the first degree is called a simple equation, an equation of the second degree which contains only one unknown letter is called a quadratic equation, or simply a quadratic, a single equation of the third degree is called a cubic equation, or simply a cubic, a single equation of the fourth degree is called a biquadratic equation, or sim

ply a biquadratic, and generally a single equation of the nth degree is said to be an equation of the nth degree.

Thus, 2x+3=7, 4x-3y= 5 are simple equations, 2+ = 4 is a quadratic, 23 3x2 + 4x = = 15 is a cubic, 4x 6x2+5x= 40 is a biquadratic, and generally, if n stands for any positive integer, the equation a+b-1 + can-2 + etc. p is said to be of the nth degree or power, noticing that the letters a, b, c, etc., may one or more of them be negative, and that one or more of the letters b, c, etc., may be 0 if required.

=

(9.) If a single algebraic equation has terms into which all the positive integral powers of the letter, from the highest to the first, enter separately as factors, at the same time that the equation has a term (called the absolute term), into which the letter does not enter (unless it is conceived to have 0 for its index, which makes it an expression for 1), then the equation is said to be complete; but if any powers of the letter are wanting in the equation, it is said to be incomplete; and it is clear that the equation is incomplete, because the coefficients of the terms which are wanting equal 0, so that the deficient terms may be supplied by writing them with 0 for each of their coefficients. Thus, the equations ax b, ax2 + bx c, ax + bx3 + cx2 + dx e are complete equations; and a2 = c, ax + bx2 = c, ax3 + cx = d are incomplete, which are completed by writing ax2+0 × x = c, ax2 + bx2 + 0 x x = c, etc., for them.

=

(10.) Equations which involve only numbers together with the unknown letter or letters are called numeral, or numerical equations, while those that involve known letters, or known letters and numbers, together with one or more unknown letters, are called literal equations. Thus, 3x-5 = −x + 7, xy = 12 are numerical equations, and ax = b, 3x2-4ax 30 are literal equations.

(11.) Because the members of an equation must equal each other, if we subtract the right member from the first, then, by Ax. 6, p. 8, the result must equal 0; consequently, if the equation is a single equation, and x the unknown letter, then denoting the result of the subtraction by X, the

equation will be reduced to X = 0, which may clearly be an algebraic or transcendental equation.

If the equation X=0 is determinate, it is clear, from what has heretofore been shown, that its real roots can be found by the rule of Double Position in Arithmetic.

(12.) It is clear that the solution of any single determinate equation consists in transforming the equation in such a way that the unknown letter shall stand alone on one side, so that the known expression on the other side will be its value.

SOLUTION OF (SIMPLE) EQUATIONS CONTAINING ONE UNKNOWN

LETTER.

CASE I

1. Any term may be carried or transposed from one member of any equation to the other, by changing its sign, as is clear from Ax. 11.

2. Any term which is common to both members of any equation, and has the same sign in each, may be erased or stricken out. If the signs of all the terms of any equation are changed, it is clear that the equation will still exist.

EXAMPLES.

1. Given, 53 and y+4= 9, to find x and y.

-

--

Adding 5 to both members of the first, and subtracting 4 from both members of the second, they will become x-5+ 535 and y+4-49-4, or since 550, 3+ 5 = 8,440, and 945, we shall get a = 8 and y = 5. Putting 8 for a and 5 for y in the given equations, they become 8-5 = 3 and 5 + 4 = 9, which being identical equations, it follows that x and y have been correctly found.

2. Given, x + 8 = 70 and y — 16 = 4, to find x and y. Ans. x 62 and y = 20.

3. Given, x+a+b=b+c and y-d-e=f-e, to find x and y.

Erasing b and e, and transposing a and d, the answers are x = c — a, y = d + f.

4. Given,3 + 2x2-3x-5=-3x2- 17x+9, to