2x+3y+4z=100

x+y+z=35

if x,y,z are at least over 1, what is the maximum and minimum of x?

jdh3010
Aug 10, 2015

#2**+10 **

2x + 3y + 4z = 100 ...(1)

x + y + z = 35 ...(2)

Multiply (2) by 2 and subtract from (1) to get y = 30 - 2z ...(3)

Multiply (2) by 3 and subtract (1) from the result to get x = 5 + z ...(4)

I'll assume you want integer solutions, and that "at least over 1" means 2 or more (you can modify the following appropriately if you mean something else).

Then, using (3), z = 2 means y = 26 (which is greater than 1, so ok), and y = 2 means z = 14, so z varies from 2 to 14.

From (4), because x is a linear function of z, we have minimum(maximum) of x corresponds to minimum(maximum) of z, so

minimum x = 5+2 = 7

maximum x = 5+14 = 19

.

Alan
Aug 10, 2015