■-'ft:'', 
INDETERMINATE PROBLEMS. 
191 
to be 
inti 
i, an- 
iftfw, 
’ thick 
— 3J5 
= 138 
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PROBLEM XI. 
If 5x + 7y + 11 z — 224; it is required to find all 
the possible values of x, y, and z, in whole numbers. 
In this, and other questions of the same kind, where 
you have three or more indeterminate quantities and 
only one equation, it will be proper, first of all, to 
find the limits of those quantities. Thus, in the pre 
sent case, because x is — — — tLlj. and because 
the least values of y and z cannot (by the question) be 
less than unity, it is plain that x cannot be greater than 
004 n 1J 
— , or 41 : and, in the same manner it will 
appear that y cannot be greater than 29, nor z greater 
than 19 ; which therefore are the required limits in this 
,, . 224 —7y — 112 
case. Moreover, since x is = ; 
5 
1 + 2 y + 
45 
nifest that 
2 y 4-2+1 
— a whole number, it is ma- 
must also be a whole number: 
let z + l be therefore considered as a known quan 
tity, and let the same be represented by b, and then 
2 y + b 
the last expression will become 
from which 
by proceeding as above, we shall get y — 2b — 22 
+ 2; whence the corresponding value of x comes out 
~ 42 — 5z. 
Let z be now taken =r 1, then will x — 37 and y 
— 4; from the former of which values, let the co-efli- 
cient of y be, continually, subtracted, aud to the latter, 
let that of x be continually added, and we shall thence 
have 37, 30, 23, 16, 9, and 2, for the successive values 
of x; and 4, 9, 14, 19, 24, and 29, for the correspond 
ing values of y: which are all the possible answers when 
z — 1. 
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