XII. SIMPLE EQUATIONS. CONTAINING TWO OR MORE UNKNOWN QUANTITIES. 189. An equation containing two or more unknown quantities is satisfied by an indefinitely great number of sets of values of these quantities. will satisfy the given equation. For this reason, an equation containing two or more unknown quantities is called an indeterminate equation. 190. Let there be two equations, each of the first degree (Art. 179), involving the same two unknown quantities, such as and 6x-5y=47, (1) (2) By Art. 189, each equation by itself is satisfied by an indefinitely great number of sets of values of x and y. But we shall find that there is but one set of values of x and y which satisfies both equations at the same time. For, multiplying each term of (1) by 4, and each term of (2) by 5, we have and 24 x 20 y 128, 35x+20 y = 70. (3) (4) Adding (3) and (4) (see Note below), we obtain Substituting this value of x in (2), we have (5) The set of values x = 2, y=7, satisfies both (1) and (2); and no other set of values of x and y can be found which will satisfy both equations at the same time. Note. We speak of adding a set of equations when we mean placing the sum of the first members equal to the sum of the second members. Abbreviations of this kind are often used in Algebra; thus we speak of multiplying an equation when we mean multiplying each of its terms. 191. A series of equations is called Simultaneous when each contains two or more unknown quantities, and all the equations of the series are satisfied by the same set or sets of values of the unknown quantities. 192. The solution of a series of simultaneous equations is the process of finding the set or sets of values of the unknown quantities involved in them, which satisfy all the equations at the same time. 193. Two simultaneous equations are said to be Independent when neither of them is satisfied by every set of values of the unknown quantities which satisfies the other. Three or more simultaneous equations are said to be independent when neither of them is satisfied by every set of values of the unknown quantities which satisfies the other equations simultaneously. 337312A Thus, the equations x+y=9 and a-y=1 are independ ent; for the second equation is not satisfied by every set of values of x and y which satisfies the first. But x+y=9 and 2x+2y= 18 are not independent; for the second equation can be made to assume the same form as the first by dividing each term by 2; and hence every set of values of x and y which satisfies the first equation also satisfies the second. 194. It is evident from Art. 190 that two independent, simultaneous equations, each of the first degree, and both involving a certain unknown quantity, can be combined so as to obtain a single equation of the first degree which does not contain that unknown quantity. Thus, in Art. 190, both (1) and (2) involve y; and by combining the equations, we obtain equation (5), which does not contain y. The unknown quantity which does not appear in the single equation is said to have been eliminated. 195. There are four principal methods of elimination. I. Elimination by Addition or Subtraction. The example in Art. 190 is an illustration of elimination by addition; we will now give an example of elimination by subtraction. Example. Solve the equations 15x+8y = 1. (1) 10x-7y=-24. (2) Multiplying (1) by 2 (Art. 190, Note), RULE. If necessary, multiply the given equations by such numbers as will make the coefficients of one of the unknown quantities in the resulting equations of equal absolute value. Add or subtract the resulting equations according as the coefficients of equal absolute value are of unlike or like sign. Note. If the coefficients which are to be made of equal absolute value are prime to each other, each may be used as the multiplier for the other equation; but if they are not prime to each other, such multipliers should be used as will produce their lowest common multiple. Thus, in Art. 190, to make the coefficients of y of equal absolute value, we multiply (1) by 4 and (2) by 5; but in the example of Art. 195, to make the coefficients of r of equal absolute value, since the L.C.M. of 10 and 15 is 30, we multiply (1) by 2 and (2) by 3. From one of the given equations find the value of one of the unknown quantities in terms of the other, and substitute this value in place of that quantity in the other equation. From each of the given equations find the value of the same unknown quantity in terms of the other, and place these values equal to each other. 198. IV. Elimination by Undetermined Multipliers. Note An Undetermined Multiplier is a multiplier, at first undetermined, but to which a convenient value is assigned in the course of the operation. |