I 
DETERMINATE EQUATIONS 485 
I. 9. a — x = m{b — x). 
I. 10. x + b = m(a—x). 
* 
I. 11. x+b = m(x — a). 
I. 39. (a + x)b + {b+x)a = 2 (a + b)x, \ 
or (a + b)x + (b + x)a = 2(a + x)b, V (a > b) 
or [a + b)x + [a + x)b = 2 (b + x)a. j 
Diophantus states this problem in this form, ‘ Given 
two numbers {a, b), to find a third number (x) such that 
the numbers 
(a + x)b, (b + x)a, (a + b)x 
are in arithmetical progression.’ 
The result is of course different according to the order 
of magnitude of the three expressions. If cob {5 and 3 
are the numbers in Diophantus), then (a + x)b < (b + x)a; 
there are consequently three alternatives, since {a + x)b 
must be either the least or the middle, and (b + x) a either 
the middle or the greatest of the three products. We may 
have 
(a + x)b < (a + b)x < {b + x) a, 
or (a + b)x < {a + x)b < (h + x)a, 
or {a + x)b < (b + x)a < {a + b)x, 
and the corresponding equations are as set out above. 
(ii) Determinate systems of equations of the first degree. 
I. 1. x + y = a, x — y = b. 
jl. 2. x + y = a, x = my, 
il. 4. x — y = a, x = my. 
I. 3. x + y = a, x = my + b. 
j I. 5. x + y = a, — x+-y = b, subject to necessary condition. 
m 
n 
I. 6. x + y = a, — x— -y=b. 
y ' m n J