144. In addition to the principles detailed in (Art. 150.) we may subjoin the following: If two equal quantities be raised to the same power, the results will be equal. If the same root of two equal quantities be extracted, the results will be equal. Hence, any equation may be cleared of a single radical quantity, by transposing all the other terms to the opposite side, and then raising each member to the power denoted by the index of the radical. If there be more than one radical, the operation must be repeated. Thus: ON THE SOLUTION OF SIMPLE EQUATIONS, CONTAINING TWO OR MORE UNKNOWN QUANTITIES. 145. A single equation, containing two unknown quantities, admits of an infinite number of solutions; for if we assign any arbitrary value to one of the unknown quantities, the equation will determine the corresponding value of the other unknown quantity. Thus, in the equation y = x + 10, each value which we may assign to a will, when augmented by 10, furnish a corresponding value of y. An equation of this nature is called an indeterminate equation, and since the value of y depends upon that of x, y is said to be a function of x. In general, every quantity, whose value depends upon one or more quantities, is said to be a FUNCTION of these quantities. Thus, in the equation y = a x + b, we say that y is a function of x, and that y is expressed in terms of x, and the known quantities a, b. If, however, we have two equations between two unknown quantities, and if these equations hold good together, then it will be seen that we can combine them in such a manner as to obtain determinate values for each of the unknown quantities. In general, in order that questions of this nature may admit of determinate solutions, we must have as many separate equations as there are unknown quantities; a groupe of equations of this nature is called a system of simultaneous equations. 146. In order to solve a system of two simple equations containing two unknown quantities, we must endeavour to deduce from them a single equation, containing only one unknown quantity; we must therefore make one of the unknown quantities disappear, or, as it is termed, we must eliminate it. The equation thus obtained, containing one unknown quantity only, will give the value of the unknown quantity which it involves, and substituting the value of this unknown quantity in either of the equations containing the two unknown quantities, we shall arrive at the value of the other unknown quantity. The process which most naturally suggests itself for the elimination of one of the unknown quantities, is to derive from one of the two equations an expression for that unknown quantity in terms of the other unknown quantity, and then substitute this expression in the other equation. We shall see that the elimination may be effected by different methods, which are more or less simple according to the nature of the question proposed. 147. FIRST METHOD.—From equation (1) we find the value of y in terms of x, which gives y = x + 6; substituting the expression x6 for y in equation (2) it becomes x + 6 + x = 12, from which we find the determinate value x= 3; since we have already seen that y = x + 6, we find also the determinate value y = 3 + 6 or 9. Thus it appears, that although each of the above equations, considered separately, admits of an infinite number of solutions, yet the system of equations admit only one common solution, x = 3, y = 9. 148. SECOND METHOD.-Derive from each equation an expression for y in terms of x, we shall then have These two values of y must be equal to one another, and, by comparing them, we shall obtain an equation involving only one unknown quantity, viz. -x, Substituting the value of x in the expression y = x + 6, we find y = 9. The substitution of 3, the value of x, in the second expression, y = 12 leads necessarily to the same value of y, for we derived the value of x from the equation + 6 = 12 x. 149. THIRD METHOD.-Since the coefficients of y are equal in the two equations, it is manifest that we may eliminate y by subtracting the two equations from each other, which gives Having thus obtained the value of x, we may deduce that of y by making x = 3 in either of the proposed equations; we can however determine the value of y directly, by observing, that, since the coefficients of a in the proposed equations are equal and have opposite signs, we may eliminate x by adding the two equations together, which give (y — x) + (y + x) = 12 + 6 Whence y=9 If we examine the three above methods, we shall perceive that they consist in expressing that the unknown quantities have the same values in both equations. These methods have derived their names from the processes employed to effect the elimination of the unknown quantities. The first is called the method of elimination by substitution. The third comparison. |
