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roots will always be real: whereas, in an equation of an evei. number of dimensions, all its roots may be imaginary ; as roots of this kind always enter into an equation by pairs.
Such are the equations a? 6x + 14 = 0, and 24 2003 9x2 + 10x + 50 = 0.*
OF THE RESOLUTION OF SIMPLE EQUATIONS,
Containing only one unknown Quantity: The resolution of simple, as well as of other equations, is the disengaging the unknown quantity, in all such expressions, from the other quantities with which it is connected, and making it stand alone, on one side of the equation, so as to be equal to such as are known on the other side; for the performing of which, several axioms and processes are required, the most useful and necessary of which are the following :-*
Any quantity, may be transposed from one side of an equation to the other, by changing its sign; and the two members, or sides, will still be equal.
Thus, if x +3=7; then will x=7 - 3, or x = 4.
* To the properties of equations abovementioned, we may here farther add:
1. That the sum of all the roots of any equation is equal to the coefficient of the second term of that equation, with the sign changed.
2. The sum of the products of every two of the roots, is equal to the coefficient of the third term, without any change in its sign.
3. The sum of the products of every three terms of the roots, is equal to the coefficient of the fourth term, with its sign changed.
4. And so on, to the last, or absolute term, which is equal to the product of all the roots, with the sign changed or not, according as the equation is of an odd or an even number of dimensions. See, for a more particular account of the general theory of equations, Vol. II. of Bonnycastle's Treatise on Algebra, 8vo., 1820; or Ryan's Elementary Treatise on Algebra, 12mo., 1824.-Ed.
+ The operations required for the purpose here mentioned, are chiefly such as are derived from the following simple and evident principles :
1. If the same quantity be added to, or subtracted from, each of two equal quantities, the results will still be equal; which is the same, in effect, as taking any quantity from one side of an equation, and placing it on the other side, with a contrary sign.
2. If all the terms of any two equal quantities, be multiplied or divided, by the same quantity, the products, or quotients thence arising, will be equal.
3. If two quantities, either simple or compound, be equal to each other, any like powers, or roots, of them will also be equal.
All of which axioms will be found sufficiently illustrated by the processes arising out of the several examples annexed to the six different cases given in the text.
And, if a 4+6=8; then will x = 8 + 4 6 = 6.
d; then will a = a
bto-d. And, if 4x 8 = 3x + 20; then 4x 3x = 20 + 8, and consequently x = 28.
From this rule it also follows, that if a quantity be found on each side of an equation, with the same sign, it may be left out of both of them; and that the signs of all of the terms of any equation may be changed from + to + without altering its value. Thus, if x + 5 = 7 + 5; then, by cancelling, x = 7. And if a - x=b-c; then, by changing the signs,
b, or x = a to b.
EXAMPLES FOR PRACTICE.
1. Given 2x + 3 = x + 17 to find x.
8=8 5 to find x.
If the unknown quantity, in any equation, be multiplied by any number, or quantity, the multiplier may be taken away, by dividing all the rest of the terms by it; and if it be divided by any number, the divisor may be taken away, by multiplying all the other terms by it.
Thus, if ax = 3ab — c; then will x = 36
And, if 2x + 4 = 16; then will x + 2 = 8,
or x = 8– 2 = 6.
5 +3; then will x = 10 +6= 16.
230 And, if 2=4; then 2x — 6= 12, or by division,
EXAMPLES FOR PRACTICE.
1. Given 16x + 2 34 to find x.
= 2. 2. Given 4ic .8 = 3x + 13 to find x. Ans. x = 3. 3. Given 10x 19 = 7x + 17 to find X. Ans, X = 12. 4. Given 8x 3 +9 7x +9+27 to find x.
Ans. x = 2
4d 5. Given 3ax 3ab = 12d.
Any equation may be cleared of fractions, by multiplying each of its terms, successively, by the denominators of those fractions, or by multiplying both sides by the product of all the denominators, or by any quantity that is a multiple of themi.
3х Thus, if
5, then, multiplying by 3, we have x + 3 : 15; and this, multiplied by 4, gives 4x + 3x = 60; whence, by addition, 7x = 60, or x =
7 And, if + = 10; then, multiplying by 12, (which is a
4 multiple of 4 and 6,) 3x + 2x = 120, or 5x 120, or x = 120
= 24. 5
It also appears, from this rule, that if the same number, or quantity, be found in each of the terms of an equation, either as a multiplier or divisor, it may be expunged from all of them, without altering the result. Thus, if ax ab t'ac; then by cancelling, x =
b And if
then, x +b=c, or x=(b.
62 to find a.
Ans. 20 = 60. 2
a +19 20
to find X. Ans. x=9. 3
x + 3 = 16
to find X. 3
Ans. * 13. a +7 + to find X. d
acd + abd – 2cbd CASE IV
If the unknown quantity, in any equation, be in the form of a surd, transpose the terms so that this may stand alone, on one side of the equation, and the remaining terms on the other
(by Case I.); then involve each of the sides to such a power as corresponds with the index of the surd, and the equation will be rendered free from any irrational expression.
Thus, if V x — 2=3; then will ✓ x= 3 + 2 = 5, or, by squaring, a
52 And if v (3x + 4) = 5; then will 3x + 4= 25, or 3x=25
21 21, or x = Also, if V (2x + 3) +4= 8; then will v (2x + 3) = 8–4 4, or 2x +3=43=64, and consequently 2x = 64 – 3=61, 61
- 4 =
EXAMPLES FOR PRACTICE,
1. Given 2 V x + 3 = 9 to find x.
Ans. x = 9. 2. Given ✓ (x + 1) - 2 = 3 to find x.
24. 3. Given (3x + 4) + 3 = 6 to find X.
Ans. x = 4. Given ✓(4 + x) = 4
✓ x to find x.
Ans. 3 =
24. 5. Given ✓ (4a2 + x) =V (464 + **) to find x.
34 Ans. 2 =
If that side of the equation which contains the unknown quantity, be a complete power, the equation may be reduced to a lower dimension, by extracting the root of the said power on both sides of the equation. Thus, if x2 = 81;
81 ; then x =V81 = 9; and if 23 = 27, then 2 = 27 = 3.
33 Also, if 3x2 9= 24; then 3x2 = 24 + 9 = 33, or a3
3 = 11, and consequently x= V11.
And, if 202 + 6x +9=27; then, since the lefthand side of the equation is a complete square, we shall have, by extracting the roots, " + 3 = V27=V(9.X 3) =3v3, or x = 3 V3 - 3.
EXAMPLES FOR PRACTICE.
1. Given 9x2 - 6= 30 to find ic.
Ans. 2 =
2. 2. Given 203 +9 : 36 to find x.
Ans. X=3 81 3. Given aca + +
to find x.
Ans. x = 4 4
a 4. Given na to ax + 73 to find a. Ans. 3 =
2 5. Given 3* + 14x +49 = 121 to find a. Ans. x = 4
CASE VI. Any analogy, or proportion, may be converted into an equation, by making the product of the two extreme terms equal to that of the two means. Thus, if 3x : 16 ::5:6; then 3. X 6 = 16 X 5, or 18x
80 40 4 80, or x =
4 18 9 9 2x
2CX And if a::b:c; then will
ab, or 2cx 3
3ab by division, x= 2c
4x Also, if 12 --- 3:
: :: 4:1; then 12
2x, or 200
12 + x = 12, and consequently x = = 4.
3ab ; or,
1. Given 5x - 15 = 2x + 6 to find the value of x. Here 5x 2x = 6 + 15, or 3x = 6 + 15 = 21; and there
21 fore x =
3. 2. Given 40 6x 16 = 120 - 14x, to find the value of x. Here 14% 6x == 120 --- 40 + 16; or 8x = 136 - 40=96;
96 and therefore x=-= 12.
8 3. Given 3x2 - 10x = 8x + 2*, to find the value of x.
Here 3x – 10 = 8 + , by dividing by x; or 3x x=8+ 10 = 18, by transposition.
18 And consequently 2x = 18, or x = : 9.
2 4. Given 6ax3 - 12abxa
12abx* = 2ax + 6uxʻ, to find the value of x. Here 2x - 46
2 x X + 2, by dividing by 3ax* ; or 2x 2 + 4b; and therefore x = 4b + 2.