Question 261355
You need to purchase 100 stamps and only spend $100.  Using only .25, $1, $15 stamps, and you need to buy at least one of each type of stamp, how many of each stamp do you buy?
<pre><font size = 4 color = "indigo"><b>
{{{system(x+y+z=100,
.25x+y+15z=100,
x>=1, y>=1,z>=1)}}}

Solve the two equations for x and y in terms of z:

{{{system(x=(56/3)z,y=100-(59/3)z, x>=1, y>=1,z>=1)}}}

So z must be a multiple of 3 in order for x and y to be integers. 

Since {{{y>=1}}},

{{{100-(59/3)z>=1}}}
{{{300-59z>=3}}}
{{{-59z>=-297}}}
{{{z<=297/59}}}
{{{z<=5&2/59}}}

{{{1 <= z <= 5}}}

And the only multiple of 3 between 1 and 5 is 3.

So z must be 3

So {{{x=(56/3)z=(56/3)*3=56}}}

and {{{y=100-(59/3)z = 100-(59/3)(3)=100-59=41}}}

So the only solution is 56 $.25 stamps, 41 $1 stamps and 3 $15 stamps.

Edwin</pre>