178 A Treatise of Algebra.

Q_U E S T I O N 11.

Is 17X -f- igy st- 2iz == 4 00 > ’Ar proposed to find all thepossible Values of x, y and z, in whole positive Numbers.

When the Coefficients of the indeterminate Quanti-ties *, y and z are nearly equal, as in this Example, it■will be convenient to substitute for the Sum of thoseQuantities: Thus, let x st- y st- z be put — m ; thenby subtracting 17 times this last Equation from the pre-ceding one, we shall have 2y-st- 42; 1= 400 —17 m; andby subtracting the given Equation from 21 times theassumed one * st- y st- z — w, there will remain4* st- 2.y=.2im- —400. Therefore, since y and Z canhave no Values less than Unity, it is plain, from the firstof these two Equations, that 400 — 17m cannot be lest

400 —6

than 6, and therefore m not greater than —- — or

J 7

23 : Also, because by the second of the two last Equa-tions, 21m —? 400 cannot be less. than. 6 ? it^lsj oh-' 40b -j- 6 „

vious that m cannot be less than — - wSio:

* ‘ . rt

'* . . -v 1

Therefore 19 and 23 are'the Limits of m, in threstUase.These being determined, let 4* be transposed-in thelast Equation, and the whole divided by 2, and we

m -

shall have y = 10m — 200 — 2* st- - ; which being a

2

whole Number, by the Question, it is evident that

— must likewise be a whole Number, and consequent-2

ly m equal to an even Number; which, as the Limitsof m are 19 and 23, can only be 20, or 22 : Let,therefore, m be first taken — 20, then y will become

— i o —2* and z (m — x — y) 1 o st- x ; wherein* beingtaken equal to i, 2, 3 and 4 successively, we shall havey equal to 8, 6, 4, 2 and z equal to I I, 12 , 13, 14respectively, which are four of the Answers required.Again, let m be taken = 22, then will y = 3 3 — 2*

and