XII. SIMULTANEOUS EQUATIONS OF THE FIRST DEGREE WITH MORE THAN TWO UNKNOWN QUANTITIES. 182. If there be three simple equations and three unknown quantities, deduce from two of the equations an equation containing only two of the unknown quantities by the rules of the preceding chapter; then deduce from the third equation and either of the former two another equation containing the same two unknown quantities; and from the two equations thus obtained the unknown quantities which they involve may be found. The third quantity may be found by substituting the above values in any of the proposed equations. For convenience of reference the equations are numbered (1), (2), and (3), and this numbering is continued as we proceed with the solution. Multiply (1) by 3 and (2) by 2; thus, Substitute the values of y and z in (1); thus, The same method may be applied to any number of simple equations. EXAMPLES OF SIMULTANEOUS EQUATIONS OF THE FIRST DEGREE WITH MORE THAN TWO UNKNOWN QUANTITIES. 1. 2. 3. 4. 5. 6. 3x + 2y - 4z=15, 5x-3y+2z=28, 3y+4z-x= 24. x + y − z = 1, 8x + 3y - 6z = 1, 3z-4x - y = 1. 9. xyz = a (yz — zx − xy) = b (zx − xy — yz) = c (xy — yz − zx). 15. 4x-3y+2x=9, 2x+5y-3x= 4, 5x+6y-2z = 18. XIII. PROBLEMS WHICH LEAD TO SIMPLE EQUATIONS WITH MORE THAN ONE UNKNOWN QUANTITY. 183. We shall now give some examples of problems which lead to simple equations with more than one unknown quantity. A and B engage in play; in the first game A wins as much as he had and four shillings more, and finds he has twice as much as B; in the second game B wins half as much as he had at first and one shilling more, and then it appears he has three times as much as A; what sum had each at first? Let x be the number of shillings which A had, and y the number of shillings which B had; then after the first game A has 2x+4 and B has y-x-4. Thus by the question Also after the second game A has 2x + 4-1/ 184. A sum of money was divided equally among a certain number of persons; had there been three more, each would have received one shilling less, and had there been two fewer, each would have received one shilling more than he did; required the number of persons, and what each received. Let x denote the number of persons, y the number of shillings which each received. Then xy is the sum divided; thus by the |