A L G E 
The fame method may be applied to any number of 
fimple equations. 
That the unknown quantities may have definite values, 
there mull be as many independent equations as unknown 
quantities. When there are more equations than unknown 
quantities, the value of any one of thefe quantities may 
be determined from different equations; and lhould the 
values, thus found, differ, the equations are incongruous; 
fhould they be the fame, one or more of the equations are 
unneceffary. When there are fewer equations than un¬ 
known quantities, one of thefe quantities cannot be found, 
but in terms which involve fome of the reft, whofe value 
may be affumed at pleafure; and in fuch cafes the num¬ 
ber of anfwers is indefinite. Thus, if x-\-y=a, x~a — 
y, and, affuming y at pleafure, we obtain a value of x, 
fuch, that x-\-y=a. 
Thefe equations muff alfo be independent, that is, not 
deducible one from another. 
Let x-\-y—a, and 2x J i -2y=2a ; this latter equation be¬ 
ing deducible from the former, involves no different fup- 
polition, or requires any thing more for its truth than that 
x-\-y=za fhould be a juft equation. 
Problems which produce Simple Equations. 
From certain quantities which are known, to inveftigate 
others which have a given relation to them, is the bufi- 
nefs of algebra. When a queftion is propofed to be re- 
folved, we muff: firft confider fully its meaning and condi¬ 
tions. Then, fubftituting for fuch unknown quantities as 
appear moft convenient, we muff: proceed as if they were 
already determined, and we. wifhed to try whether they 
would anfwer all the propofed conditions or not, till as 
many independent equations arife as we have affumed un r 
known quantities, which will always be the cafe if the 
queftion be properly limited; and, by the folution of thefe 
equations, the quantities fought will be determined. 
Prob. i . A bankrupt owes A twice as much as he owes 
B, and C as much as he owes A and B together; out of 300I. 
which is to be divided amongft them, what muff; each 
receive f 
Let w reprefent what B muff: receive; 
Then 2x is equal to what A muff: receive, 
And x-\-2X, or 3*, is equal to what C muff: receive. 
Amongft them they receive 300I. therefore, 
x -J- 2 x -|~ 3 x =3 o o 
6^=300 
300 
x~ -= 50, what B muff: receive. 
6 J 
2*=ioo, what A muff: receive. 
3v=i50, what Cmuft receive. 
Prob. 2. A workman was employed for fixty days, on 
condition that for every day he worked he fhould receive 
15 pence, and for every day he played he fhould forfeit 
3 pence ; at the end of the time he had 20s. to receive; 
required the number of days he worked. 
Let w be the number of days he worked; 
Then 60— x is the number he played. 
i$x his pay, in pence. 
300— $x, firm forfeited. 
15X —3oo+5a-=: 240, by the queftion. 
2o.*=24o+3 00=540 
#=27, the days he worked. 
60—x=33, the days he played. 
Prob. 3. A and B play at bowls, and A bets B three 
fhillings to two upon every game; after a certain number 
of games it appears that A has won three fhillings, but 
had he ventured to bet five fhillings to two, and loft one 
game more out of the fame number, he would have loft 
thirty fhillings: how many games did they play ? 
Let x be the number of games A won; 
y the number B won. 
Then 2x is what A won of B, 
And 37 what B won of A. 
~2x —37=3, by the queftion; 
Vol. I. No. 19. 
BRA. 
289 
x —1.2 A would win on the 2d fuppofition. 
y- j-1.5 B would win. 
574 - 5 —^2*-f2=30 
$y— 2X=20 —5—2=23 
Therefore, ax —372=3 
And 57—2^=23 
By addition, 57—347=26 
27=26 
y —13 
2^=3+37=3+39=42 
' A'=2I 
x-\-y=z 34; the number of games. 
On Quadratic Equations. 
When the terms of an equation involve the fquare of 
the unknown quantity, but the firft power does not ap¬ 
pear, the value of the fquare is obtained by the preceding 
rules; and, by ex' radiing the fquare root on both Tides, the 
quantity itfelf is found. 
Ex. Let 5„r 2 —45=0; to find x. 
By tranf. 5^=45 
A'~—Ttj 
Therefore, x—\/c)—± 2 - 
The figns and — are both prefixed to the root, be- 
caufe the fquare root of a quantity may be either pofitive 
or negative. The fign of x may alfo be negative, but 
ftill, x will be either equal to -j-3 or —3. 
Ex. Let ax"~bcd\ to find a-. 
1 bed 
a 
V \bcd 
X2=±'Sj -- 
a 
If both the firft and fecond powers of the unknown 
quantity be found in an equation, arrange the terms ac¬ 
cording to tfie dimenfions of the unknown quantity, be¬ 
ginning with the higheft, and tranfpofe the known quanti¬ 
ties to the other fide ; then, if the fquare of the unknown 
quantity be affedted with a coefficient, divide all the terms 
by this coefficient, and, if its fign be negative, change the 
figns of all the terms, that the. equation may be reduced 
to this form, x z dzpx~dzq. Then add to both Tides the 
fquare of half the coefficient of the firft power of the un¬ 
known quantity, by which means the firft fide of the equa¬ 
tion is made a complete fquare, and the other confifts of 
known quantities; and, by extradling the fquare root on 
both tides, a fimple equation is produced, from which the 
value -of the unknown quantity may be found. 
Ex. 1. Let 2r = 4-/»w=y; now weknow.that.Y 2 -j-/>w+— is 
4 
the fquare of *+-; add therefore ~ to both Tides. 
2 4 
and we have 
p 2 P*‘ 
x -{-px-\ ——— q-\ - 
4 4 
root on both Tides, 
2 • 4 
—-±V ?+—• 
2 4 
In t he fam e manner, if x 2 
P -±\] q+~. 
2 4 
Ex. 2. Let x 2 —i2A’-|-35=:o; tofindw. 
By tranf. x 2 —.122=—35, and, adding the fquare of 
6 to both lides of the equation, 
x 2 —i2.e+36=36—35=1 ; then, extraft- 
ing the fquare root on both tides, 
x —6=±:i 
And a-=6±i= 7, 01-5; either of which, fub- 
3 E ftituted 
then, by extracting the fquare 
and by tranfpofttion, 
~px—q , is found to be 
