Here we subtract (1) from (2), and obtain (6); then (6) from (3), and obtain (7); then (7) from (4), and obtain (8); then (8) from (5), and obtain (9), which reduced gives (10), or x = 9. Substituting this value of x in (1), (6), (7), and (8), and reducing, we obtain (11), (12), (13), and (14), or y = 17, z= 12, w = u = 37. 44, and Hence, for solving equations containing any number of unknown quantities, RULE. From the given equations deduce equations one less in number, containing one less unknown quantity; and continue thus to eliminate one unknown quantity after another, until one equation is obtained containing but one unknown quantity. Reduce this last equation so as to find the value of this unknown quantity; then substitute this value in an equation containing this and but one other unknown quantity, and reducing the resulting equation, find the value of this second unknown quantity; substitute again these values in an equation containing no more than these two and one other unknown quantity, and reduce as before; and so continue, till the value of each unknown quantity is found. NOTE. The process can often be very much abridged by the exercise of judgment in selecting the quantity to be eliminated, the equations from which the other equations are to be deduced, the method of elimination which shall be used, and the simplest equations in which to substitute the values of the quantities which have been found. Find the values of the unknown quantities in the following equations : 20; and if NOTE. - If these equations are added together and the sum divided by 4, we shall have x + y + z + w + u = from this the given equations are successively subtracted, the values of the unknown quantities become known. r x = 2. PROBLEMS PRODUCING EQUATIONS OF THE FIRST DEGREE CONTAINING MORE THAN TWO UNKNOWN QUANTITIES. 118. 1. A merchant has three kinds of flour. He can sell 1 bbl. of the first, 2 of the second, and 3 of the third for $85; 2 of the first, 1 of the second, and bbl. of the third for $45.50; and 1 of each kind for $41. What is the price per bbl. of each? Ans. 1st, $12; 2d, $14; 3d, $15. 2. Three boys, A, B, and C, divided a sum of money among themselves in such a manner that A and B received 18 cents, B and C 14 cents, and A and C 16. How much did each receive? Ans. A, 10; B, 8; C, 6 cents. 3. As three persons, A, B, and C, were talking of their ages, it was found that the sum of one half of A's age, one third of B's, and one fourth of C's was 33; that the sum of A's and B's was 13 more than C's age; while the sum of B's and C's was 3 less than twice A's age. What was the age of each? Ans. A's, 32; B's, 21; C's, 40. 4. As three drovers were talking of their sheep, says A to B, If you will give me 10 of yours, and C one fourth of his, I shall have 6 more than C now has." Says B to C, "If you will give me 25 of yours, and A one fifth of his, I shall have 8 more than both of you will have left." Says C to A and B, "If one of you will give me 10, and the other 9, I shall have just as many as both of you will have left." How many did each have? 5. Divide 32 into four such parts that if the first part is increased by 3, the second diminished by 3, the third multiplied by 3, and the fourth divided by 3, the sum, difference, product, and quotient shall all be equal. Ans. 3, 9, 2, and 18. in 8 6. If A and B can perform a piece of work together days, B and C in 9 days, and A and C in 82 days, in how many days can each do it alone? Ans. A in 15, B in 18, and C in 21 days. 7. Find three numbers such that one half of the first, one third of the second, and one fourth of the third shall together be 56; one third of the first, one fourth of the second, and one fifth of the third, 43; one fourth of the first, one fifth of the second, and one sixth of the third, 35. 8. The sum of the three figures of a certain number is 12; the sum of the last two figures is double the first; and if 297 is added to the number, the order of its figures will be inverted. What is the number? Ans. 417. 9. A man sold his horse, carriage, and harness for $450. For the horse he received $25 less than five times what he received for the harness; and one third of what he received for the horse was equal to what he received for the harness plus one seventh of what he received for the carriage. What did he receive for each? Ans. Horse, $225; carriage, $175; harness, $50. 10. A man owned three horses, and a saddle which was worth $45. If the saddle is put on the first horse, the value of both will be $30 less than the value of the second; if the saddle is put on the second horse, the value of both will be $55 less than the value of the third; and if the saddle is put on the third horse, the value of both will be equal to twice the value of the second minus $10 more than one fifth of the value of the first. What is the value of each horse? Ans. 1st, $100; 2d, $175; 3d, $275. 11. The sum of the numerators of two fractions is 7, and the sum of their denominators 16; moreover the sum of the numerator and denominator of the first is equal |