ON SIMPLE ERUATIONS.
If one quantity be equal to another, or to and this equality be expressed algebraically, it constitutes an equation. Thus, x—a =b—x is an equation, of which x—a forms one side, and b — x the other.
When an equation is cleared of frac tions and surds, Wit contain the first power only of an unknown quantity, it is call ed a simple equation, or an equation of one dimension : if the square of the unknown quantity be in any term, it is called a quadratic, or an equation of two dimen sions ; and in general, if the index of the highest power of the unknown quantity be n, it is called an equation of n dimen sions.
In any equation quantities may be trans posed from one side to the other, if their signs be changed, and the two sides will still be equal.
Let x+10 =15, then by subtracting 10 from each side, x+ 10 — 10 = 15 — 10 or x = 15 — 10.
Let x-4=6, by adding 4to each side, x — 4 + 4=6 + 4, or x,-._-6+4.
If x — a + b.=y; adding a — b to each. side, x—a+ b+ a— b = y+ 3,-6; or x y + a — b.
Hence, if the signs of all the terms on each side be changed, the two sides will still be equal.
Let x— a=b — 2 x ; by transposition, — 6+2 x=—x+ a ; or a — x=2 x—b. If every term, on each side, be multiplied by the same quantity, the results will be equal.
An equation may be cleared of frac tions, by multiplying every term, succes sively, by the denominators of those frac tions, excepting those terms in which the denominators are found.
5 x Let 3 x + — = 34 ; multiplying by 4, 4 L2 x-J- 5 x-=- 136, or 17 x=136.
If each side of an equation be divided by he same quantity, the results will be equal. Let 17 x = 136; then x = 8.
'leach side of an equation be raised to the !awe power, the results -will be equal.
Let ; then x=9 x 9 = 81. Also, if the same root be extracted on nath sides, the results will be equal.
Let x = 81 ; then = 9.
To find the value of an unknown quantity n a simple equation.
Let the equation first be eleared.of frac ions, then transpose all the terms which nvolve the unknown quantity to one side tf the equation, and the known quantities o the other ; divide both sides by the ;fficient, or sum of the co-efficients, of the unknown quantity, and the value re ed is obtained.
Ex. 1. To find the value of x in tbe equation 3.r--5.---23—ce.
by transp. 3.r+.r=23-1-5 or 4z•=28.
28 by division .r=-,T=7.
F.x.2.Let s+ 2 3 2x Milt. by 20 and 2 .r x 3 Mult. by 3, and 6.r+3.r-2 sr---24s-102 by transp. 6 "A-3 T-2 x-24 —L12 or-17 x = —102 17x-102 x = 102 =6.
17 1 h Ex. 3. --1-•—=c.
a x• , , b a -1---=e a a=c a a.
s—ca .r= --b d or c a J.—x=6 a e. c7=-17x..--b a b a c a-1.
x-f- 4 Ex. 4. 5---= - -11 55—x-4=11 a--,33. 55-4-4-33=--11.:+x 84=12x 84 , s=17=` .r— 5 =12— 2 — 4 Ex. S. x-1 2 3 • 4.r — 8 2 x-I-3 .r-5= 24 6x-1-9 x-15=72-4a•-1-8 6 x-F9 x-1-4 x=72 -1-84-15 19 x=95 95 =-x 5— = 19 If there be two independent simple equations involving two unknown quanti ties, they may be reduced to one which involves only one of the unknown quan tities, by any ofthe following methods: 1st Method. In either equation find the value of one of the unknown quanti ties in terms of the other and known quantities, and for it substitute this value nt the other equation, which will then only contain one unknown quantity,wb ()cc value may be found by the rules before laid down.
Let cx-1-y=10 To find x and y From the first equat. x=10—y; hence, 2 x=20--2 y, by subst. 20-2 y--3 y= 5 20--5=-.2y 1- 3 y 15=5 y 15 _„ — 5 hence also, x=10--yr.-..--10-3--7.
2c1 Method. Find an expression for one of the unknown quantities in each qua tion ; put these expressions equal to each other, and from the resulting equation the other unknown quantity- may be found.
Let /x+Y=a To find x and p.
bx-f-cy=de From the first equat. x=a—y from the second, b x= d e--c y, and dt—c y = b de—Cy therefore rz--y— b a—b y-=-de—c y c N---b y=d e—b a c--b e—b ad e--b n Also, x=-....tt—y; that is, de—ba ca—ba—de- f-bn x =, a— c--b c—b c a—d e c—b 3d Metbod. If either of the unknovm quantities have the same co-efficient in both equations, it may be exterminated by subtracting, or adding., the equations, ac cording as the sign ofthe unknown quan tity, in the two cases, is the same or dif ferent.