e method may be applied to any number of simple

OF SIMULTANEOUS EQUATIONS OF THE FIRST DEGREE TH MORE THAN TWO UNKNOWN QUANTITIES.

2y-42-15, 5x-3y+2x=28, 3y+4z-x=24.

By+2z = 9, 2x+5y-3x=4, 5x+6y-2z = 18.

23. xyz

24.

· = a (yz — zx — xy) = b (zx − xy — yz) = c (xy — yz − zx).

x + y + z = a+b+c,

bx + cy + az = cx + ay + bz = a2 + b2 + c2.

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

T. A.

7

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 У 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,

2x+4=2(y-x-4)= 2y-2x-8;

Also after the second game A has 2x+4

4-2-1, and B has

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

185.

x = 2y+2=12.

What fraction is that which becomes equal to when its numerator is increased by 6, and equal to when its denominator is diminished by 2?

Let x denote the numerator and y the denominator of the fraction; then by the question,

Clear the first equation of fractions by multiplying by 4y;

Clear the second equation of fractions by multiplying by 2(y-2); thus,