108. Simultaneous Equations with Three Unknown Quantities. Three simultaneous equations of the first degree, containing the three unknown quantities x, y, and z, can be solved in the following manner: Multiply the first and second of the given equations by such quantities that, in the resulting equations, the coefficients of one of the unknown quantities, z suppose, may be equal; then by addition or subtraction we eliminate z. Then take the first and third, or the second and third of the given equations, and eliminate z in a similar manWe thus obtain two simultaneous equations containing the two unknown quantities x and y; and these can be solved as in Art. 104. ner. From (i.) and (ii.) we obtain x 3, y=. Substitute these values in the first of the given equations; then 6 +2 + 2 = 7; therefore z = 1. Thus x = 3, y= 1, and z = - 1 are the values required. CHAPTER IX. PROBLEMS. 109. We shall now give examples of problems which involve more than one unknown quantity, and in which the relations between the known and unknown quantities are expressed algebraically by means of equations of the first degree. Many of the problems given in Chapter VII: really contain two unknown quantities, but the given relations are in those cases of so simple a nature that it is easy to find an equation giving one of the unknown quantities in terms of the known quantities, and when one of the unknown quantities is found, the other is immediately determined. Ex. 1. Find two numbers such that the greater exceeds twice the less by three, and that twice the greater exceeds the less by 27. Let x and y be the numbers, of which x is the greater. Then we have by the conditions of the problem Now subtract the members of this last equation from the corresponding members of the second equation, and we have 3 y = 21, or y = 7. Then from the first equation x = 3+2y=3+ 14 = 17. Thus the numbers are 17 and 7. Ex. 2. A number of two digits is equal to seven times the sum of its digits, and the digit in the ten's place is greater by four than the digit in the unit's place. What is the number? Let x be the digit in the ten's place, and y be the digit in the unit's place. Then the number is equal to 10x+y, for the x represents so many tens; also the sum of the digits is x + y. Ex. 3. Find the fraction which is equal to when its numerator is increased by unity, and is equal to when its denominator is increased by unity. the denominator. Let x = x + 1 1 and = x 1 Multiply the second equation by y + 1; then x = } (y + 1). |