<PRE><B>Question 1104904</B>
&lt;br&gt;
Let x, y, and z be the numbers of $9, $2, and $0.50 pads sold.  Then
{{{x+y+z = 90}}}  [the total number of pads sold was 90]
{{{9x+2y+.5z = 90}}}  [the total cost of the pads sold was $90]&lt;br&gt;
This is a Diophantine equation -- three unknowns but only two equations; but we can find a solution using the fact that the unknowns have positive integer values.&lt;br&gt;
The general technique for solving a Diophantine system with three variables and two equations is to eliminate one of the variables to obtain a single equation in two variables; then analyze that equation to find integer solutions.&lt;br&gt;
You can choose any one of the variables to be the one to be eliminated.  The amount of work required to finish the problem can vary a great deal depending on which variable you eliminate first.  Unfortunately, it takes a huge amount of experience with solving this kind of problem to know ahead of time which variable is the best one to eliminate.&lt;br&gt;
So I will choose to eliminate variable z -- mostly because of the fractional coefficient on z in the second equation.  So double the second equation and then subtract one equation from the other:&lt;br&gt;
{{{x+y+z = 90}}}
{{{18x+4y+z = 180}}}
{{{17x+3y = 90}}}&lt;br&gt;
Quick inspection of that equation shows that one solution is (x,y) = (0,30).  However, since the problem implies that some of each size pad were sold, we can reject that solution.&lt;br&gt;
To find other solutions in positive integers, since the coefficients 3 and 17 are relatively prime, we can take our rejected (0,30) solution and add 3 to x and subtract 17 from y, giving us (x,y) = (3,13).&lt;br&gt;
The next possible solution would be found by again adding 3 to x and subtractin 17 from y; but that would give us a negative value for y.&lt;br&gt;
So our only possible solution is (x,y) = (3,13).&lt;br&gt;
That gives us z = (90-(3+13)) = 74.&lt;br&gt;
So the store sold 3 $9 pads, 13 $2 pads, and 74 50-cent pads.&lt;br&gt;
CHECK: The total cost is {{{3(9)+13(2)+74(.5) = 27+26+37 = 90}}}</PRE>