Page:Encyclopædia Britannica, Ninth Edition, v. 1.djvu/577

Rh SIMPLE EQUATIONS.] ALGEBRA 539 Here equal quantities are divided by the same quantity, and therefore the quotients are equal. 57. Rule 3. If any term of an equation be a fraction, its denominator may be taken away, by multiplying all the other terms of the equation by that denominator. If Then If Then If &quot;We have x = 35 a x = ab- ac + ad. a - - = c , x ax-b = cx and x = + x 1 b = x , Jet? + x 2 = b + x , In these examples, equal quantities are multiplied by the same quantity, and therefore the products are equal. 58. Ride 4. If the unknown quantity is found in any term which is a surd, let that surd be made to stand alone on one side of the equation, and the remaining terms on the opposite side; then involve each side to a power denoted by the index of the surd, and thus the unknown quantity shall be freed from the surd expression. Then, by transposition, ^-=10-6 = 4; And, squaring both sides, Jx x ^# = 4x4, Or ar=16. Also, if By trans. And, squaring Hence 59. Rule 5. If the side of the equation which contains the unknown quantity be a perfect power, the equation may be reduced to another of a lower order, by extracting the root of that power out of each side of the equation. Thus, if z 3 =G4a 3 , Then, by extracting the cube root, x 4a ; And if (a + xf = i 2 a 2, Then a + x = ^W- a 2 . GO. In these examples we have been able to determine the value of the unknown quantity by the rules already delivered, because in every case the first, or at most the second power of that quantity, has been made to stand alone on one side of the equation, while the other con sisted only of known quantities ; but the same methods of reduction serve to bring equations of all degrees to a pro- per form for solution. Thus, if 1-p+g+r x+l =1 -p-x+ - by proper reduction, we have x 3 + px 1 + qx = r, a cubic equation which may be resolved by rules to be afterwards explained. SECT. VII. REDUCTION OF EQUATIONS INVOLVING MOKE THAN ONE UNKNOWN QUANTITY. Gl. Having shoAvn in the last section in what manner an equation involving one unknown quantity may be resolved, or at least fitted for a final solution, we are next to explain the methods by which two or more equations, involving as many unknown quantities, may at last be reduced to one equation and one unknown quantity. As the unknown quantities may be combined together in very different ways, so as to constitute an equation, the methods most proper for their elimination must therefore be various. The three following, however, are of general application, and the last of them may be used with advantage, not only when the unknown quantity to be eliminated rises to the same power in all the equations, but also when the equations contain different powers of that quantity. G2. Metliod 1. Observe which of the unknown quantities is the least involved, and let its value be found from each equation, by the rules of last section. Let the values thus found be put equal to each other, and hence new equations will arise, from which that quan tity is wholly excluded. Let this operation be now re peated with these equations, thus eliminating the unknown quantities one by one, till at last an equation be found which contains only one unknown quantity. Ex. Let it be required to determine x and y from these two equations. 10. 23 -3y From the first equation, From the second equation, 2 10+2y 5 Let these values of x be now put equal to each other, 10 + 2y 23 -3y And we have - = - &amp;gt; o A Or And And since x = 23-; or x = W + Zy from cither of these values we find x = 4. G3. MetJiod 2. Let the value of the unknown quantity which is to be eliminated be found from that equation wherein it is least involved. Let this value and its powers be substituted for that quantity, and its respec tive powers in the other equations ; and with the new equations thus arising, let the operation be repeated till there remain only one equation and one unknown quantity. Ex. Let the given equations, as in last method, be 5x-2//=lO. From the first equation, x = - And this value equation, we have 5 x Or 115 of x being substituted in the second 23 ~% o,._in 23 -3y = 4, as before. G4. Method 3. Let the given equations be multiplied or divided by such numbers or quantities, whether known or unknown, that the term which involves the highest power of the unknown quantity may be the same in each equation. Then, by adding or subtracting the equations, as occa sion may require, that term will vanish, and a new equa tion emerge, wherein the number of dimensions of the unknown quantity in some cases, and in others the number of unknown quantities will be diminished ; and by a repeti tion of the same or similar operations, a final equation may be at last obtained, involving only one unknown quantity. EJL. Let the same example be taken, as in the illustra tion of the former methods, namely,