SONS IN ALGEBRA. 



19 



EXAMPLE. A drover bought on equal number of sheep and 



cows for 840 crowns. Ho paid 2 crowns a-heod for tho sheep, 



orowna a-head for the cows. How many did ho buy of 



each? 



Hero, let * = the number bought of each ; 



2* = the coat of the sheep in crowns ; 



And 12x = the cost of tho cows in crowns. 



Hence, Zx + 12a> = 840 by tho conditions of tho question. 



Therefore, 14 = 840 by addition j 



And = 60, tho number bought of each. 



tho last expression is obtained from the preceding one 

 by dividing each member by 14, tho co-efficient of 14*. 



It will be perceived, in this example, that the unknown 

 quantity, or number taught, is represented by the letter *; and 

 from the conditions of the problem, we obtain the quantity 

 140, which is equal to the given quantity 840 crowns. This 

 whole algebraic expression, 14a> = 840 crowns, is called an 

 equation. 



15'2. An EQUATION, tlierefore, is a proposition expressing in 

 algebraic characters the equality between one quantity or set of 

 quantities and another, or between different expressions for the 

 same quantity. 



This equality is denoted by the sign = , which is read " is 

 equal to." Thus x + a = b + c, and 5 + 8 = 17 4, are 

 equations, in one of which the sum of x and a is equal to the 

 sum of b and c ; and in the other, tho sum of 5 and 8 is equal to 

 the difference of 17 and 4. 



The quantities on the two sides of the sign = are called 

 members of the equation; the several terms on the left con- 

 stituting the first member, and those on the right the second 

 member. 



When the unknown quantity is of the first power, the proposi- 

 tion is called a simple equation, or an equation of tho first degree. 



153. The reduction of an equation consists in bringing the un- 

 known quantity by itself to one side of the sign of equality, and all 

 the known quantities to the oilier side, without destroying the 

 equality of the members. 



To effect this, it is evident that one of the members must be 

 as much increased or diminished as the other. If a quantity be 

 added to one, and not to the other, the equality will be de- 

 stroyed. But the members will remain equal 



(1.) If the same or equal quantities be added to each. Ax. 1. 



(2.) If the same or equal quantities be subtracted from each. 

 Ax. 2. 



(3.) If each be multiplied by the same or equel quantities. 

 Ax. 3. 



(4.) If each be divided by the same or equal quantities. Ax. 4. 



The principal reductions in simple equations are those which 

 are effected by transposition, multiplication, and division. 



REDUCTION OF EQUATIONS BY TRANSPOSITION. 



In the equation x 7 = 9, the number 7 being connected 

 with tho unknown quantity x by the sign , the one is sub- 

 tracted from the other. To reduce the equation, let the 7 be 

 added to both sides. It then becomes x 7+7=9 + 7. 



The equality of tho members hero is preserved, because one 

 is increased as much as tho other. But on one side we have 

 7 and + 7. As these are equal, and have contrary signs, they 

 balance each other, and may bo cancelled. The equation will 

 then be a; = 9 + 7. 



Here the value of x is found. It is shown to be equal to 

 9 + 7, that is, to 16. The equation is therefore reduced. The 

 unknown quantity is on one side by itself, and all the known 

 quantities on the other side. 



In the same manner, if x b = a ; 



Adding b to both sides, we have x 6 + 6 = a + 6 ; 



And cancelling as before, we have x a + b. Ans. 



154. When known quantities, therefore, arc connected with the 

 unknown quantity by the sign + or , the equation is reduced 

 by TRANSPOSING the known quantities to the other side, and pre- 

 fixing the contrary sign. 



This is called reducing an equation by addition or subtraction, 

 because it is, in effect, adding or subtracting certain quantities 

 to or from each of tho members. 



EXAMPLE. Reduce the equation a + 36 m = h d. 



Here, transposing + 36, we have x m = h d 36; 



And transposing m, = h d 36 + m. Ans. 



155. Whoa several terms on the same aide of an equation an 

 alike, they must be united in one, by the role* for redaction in 



addition. 



EXAMPLE. Reduce the equation a -f 5* ifc = 76. 



Here, transposing 56 4A, w hare m = 76 56+4A; 



And uniting 76 56 in one term, we hare * = 26 + 4A. Ant. 



156. The unknown quantity most also be transposed, when- 

 ever it is on both aides of the equation. It is not "ft*^"'* 1 on 

 which side it is finally placed, though it i* generally brought to 

 the left-hand aide. 



EXAMPLE. Reduce the equation 



Here, by transposition, we have 2h h d= 3x 2x j 



And by incorporation [Art. 155] h d = x. An*. 



157. When the same term, with the same sign, is on opposite 

 sides of tho equation, instead of transposing, we may expunge it 

 from each. For this is only subtracting the same quantity from 

 equal quantities. 



EXAMPLE. Reduce the equation x+3h+d = b+3h+7d, 

 Here, by expunging 3h, wo have x + d = b + Id ; 



And by transposition and incorporation x = b + 6d. Ant. 



158. As all the terms of an equation may be transposed, or 

 supposed to be transposed, and it is immaterial which member 

 is written first, it is evident that the siyns of all the terms may 

 be changed, on both sides, without affecting tho equality. 



Thus, if we have x 6 = d a ; 



Then by transposition, we have d + a = * + 6. 



Or, by changing the places of the members, z + 6 = d + a, 



159. If all the terms on one side of an equation be transposed 

 each member will be equal to 0. 



Thus, if x + 6 = d, then it is evident that x + 6 d = 0. 



EXERCISE 25. 



1. Reduce a-r2z-8 = b-4 + z + a. 



2. Reduce y + ob Tim = a + 2y db + hm. 



3. Reduce h+30 + 7* = 8-6?v + 6r d + b. 



4. Reduce bh + 21 - 4* + d = 12 - 3* + d - 71*. 



5. Reduce 5x + 10 + a = 25 + 4i + a. 



6. Reduce 5c -I- 2* -I- 12 - 3 = x + 20 + 5c. 



7. Reduce a + b-3i' = 20 + a-4x + b. 



8. Reduce x + 3-2*-4 = 34 + 3je-4-&r. 



9. Reduce 4x 2 + 18 = 5* + 8. 



10. Reduce 21-2r = 3x-8 + 2. 



11. Reduce 3 + 5* - 18 = 6* - 22. 



12. Reduce 10* + 60 + 7* = 28* + 64 - 12*. 



13. Reduce y-10-b = 6-b. 



14. Reduce x - 10 + c - 14 - c = 0. 



KEY TO EXERCISES IN LESSONS IN ALGEBBA. 

 EXERCISE 23. 



5dr 



2. ?. 



10- 



Sax 



+ 1-* 



"2rf*~ 



Saby 

 a-b* 



MM 



*_+v 



9. I 



EXERCISE 24. 

 10. Sex. 



12. 

 13. 



! 19. 



o* ajr 



2i + xy + bx 

 , 

 15. Uf. 



8. to'--l. 24. 

 - - 6 



17. 



18. 



V' -3y + 2" 



a-o 



