288 A L G E 
In extra< 5 ling the cube root of a decimal, care muft be 
taken that the decimal places be three, or fome multiple 
of three, before the operation is begun; becaufe there 
are three times as many decimal places in the cube as there 
are in the root. 
On Simple Equations. 
If One quantity be equal to another, or to nothing, and 
this equality be expreffed algebraically, it conftitutes an 
equation. Thus x—a—b—x is an equation, of which x — a 
forms one fide and b— xthe other. 
An equation, in any of whole terms the firft potver of 
one unknown quantity only is contained, is called a funple 
equation , or an equation of one dimenfion ; if the Jquart of 
an unknown quantity be in any way involved, it is called a 
quadratic, or equation of two dimenfions; and, in general, 
if the index of the higheft power of the unknown quantity 
be n, it is called an equation of n dimenfions. 
In any equation, quantities may be tranfpofed from one 
fide to the other, if their figns be changed, and the two 
Tides will Hill be equal. Let 2+10=15, then by fub- 
trafting 10 from both lides, x-\-io —10=15—10, or a= 
15—10. 
Let x —4=6, by adding 4 to both Tides, x —4+4=6 + 
4, or *=6+4. 
If x — afb—y, adding a—b to both lides, x — a-\-b^-a—b 
~yfa — b, or xz=.y-\-a- — b. 
Cor. Hence, if the ligns of all the terms on each fide 
be changed, the two Tides will Hill be equal. Let x—a~ 
b —zx; bytranfpofition, — b-\-2xz =.—x+a; ora—x=2X— b. 
If every term on each fide be multiplied by the fame 
quantity, the refults will be equal. 
Cor. An equation may be cleared of fractions by mul¬ 
tiplying every term, fucceffively, by the denominators of 
C.X 
thofe fractions. Let 3x4-——341 multiplying by 4, 12X 
4 
+5A=i36. 
An equation may be cleared of fractions at once, by 
multiplying both Tides by the produft of all the denomina¬ 
tors, or by any quantity which is a multiple of them all. 
let *4-+-=13 ; multiplying by 2X3X4. 3X4XV+2 
2 3 4 - 
X 4 X* 4 z X 3 X- v = 2 X 3 X 4 X * 3 > or 12.2+82+62=312 ; 
that is, 262=312. 
If both lides be multiplied by 12, which is a multiple of 
, . .... I 2 A . I 2 A I 2 X 
2, 3, and 4, the equation will become-1-1-=2 
3 3 4 
156, or 62+42+32= l S& 5 that is, 13^2=156. 
If each fide of an equation be divided by the fame 
quantity, the refults will be equal. Let 172=136, then 
^136_ 
~ 17 ' 
If each fide of an equation be raifcd to the fame power, 
■I 
the refults will be equal. Let 2^=9, then 2=9X9=81. 
Alfo, if the fame root be extracted on both lides, the 
refults will be equal. Let 2=81, then 2^=9. 
To find the value of the unknown quantity in a fimple 
equation, let the equation firft be cleared of fractions, then 
tranfpofe all the terms which involve the unknown quanti¬ 
ty to one fide of the equation, and the known quantities to 
the other; divide both Tides by the coefficient, or fum of 
the coefficients, of the unknown quantity, and the value 
required is obtained. 
Ex. To find the value of 2 in the equation 3*—5= 
53— 
By tranfp. 32+2=23+5 
Or 42=23 
28 
By divifion x=—=7 
4 
If there be two independent fimple equations involving 
two unknown quantities, they may be reduced to one 
which involves only one of the unknown quantities, by 
either of the following methods: 
B R A, 
Firft, In either equation find the value of one of the 
unknown quantities in terms of the other and known quan¬ 
tities, and for it fubftitute this value in the other equation 
which will then only contain one unknown quantity, whofc 
value may be found by the rules before laid down. 
Let 2+41=10 
And 2.v—341=5 
From the firft equat. 2=10—4/ and 22=20_2 y 
By fub. 20—2 y —3^=5 
20—5=2 7+37 
15=57 
- 7 - 
Hence alfo 2=10— -y—to —3=7. 
Secondly, If either of the unknown quantities have the 
fame coefficient in both equations, it may be exterminated 
by fubtrafling or adding the equations according as the 
fign of the unknown quantity, in the two cafes” is the 
fame or different. 
Let 
f 2+4=15 \ 
\x— >=7 / 
To find x and 4/. 
By fubtraffion, 24=8, and 4=4 
Byaddition, 22=22, and a— ir. 
If the coefficients of the unknown quantity to be exter¬ 
minated be different, multiply the terms of the firft equa¬ 
tion by the coefficient of that unknown quantity in the Se¬ 
cond, and the terms of the fecond equation by its coeffici¬ 
ent in the firft, then add or fubtrafl the refulting equations 
as in the former cafe. 
Let {22+^81} To find 2 and 4,. 
Multiply the terms of the firft equation by 2, and the 
terms of the other by 3, 
Then 6a— 104= 26 
62+217=243 
By fub fraction, —317=—217 
And 4=- —7 
3 1 
Alfo 3 a— 54=13 or 3 a— 35=13 
Therefore 32=1 34-35=48 
And a=A—j6. 
3 
If there be three independent fimple equations, and 
three unknown quantities, reduce two of the equations 
to one, containing only two of the unknown quantities, by 
the preceding rules; then reduce the third equation and 
either of the former to one, containing the fame two un¬ 
known quantities; and, from the two equations thus ob¬ 
tained, the unknown quantities which they involve may be 
found. The third quantity may be found by fubftituting 
their values in any of the propofed equations. 
f 2 *-}- 37 + 4 =i 6 ') 
Ex. Let < 32+24/—52=8 J. To find a, y, and z, 
15*—67+3=6 J 
From the two firit equat. 6x-\-qy-\-i2z—^ 
6 a 4~ 47—102=16 
By fubtr. 54/4-222=32 
From the firft and third, 102-4157-4202=80 
ioa—124+62=14 
By fubtr. 274/+142=68 
And 54+222=32 
Hence 1354/4-702=2340 
And 1357+5942=864 
By fubtr. 5242=524 
2=1 
541+222=32 
That is, 54/4-22=32 
54=32—22=10 
10 
^ 5 ~ 2 
2X-]-3y-\-4.z—i6 
That is, 22+6+4=16 
2 a= 16—6—4=6 
A= 3 «. 
The 
