Question 1105118
Let x, y, and z be the numbers of $9, $2, and $0.50 pads sold, respectively.  Then<br>
{{{x+y+z = 90}}}  a total of 90 pads were sold
{{{9x+2y+.5z = 90}}}  the total cost of the pads sold was $90<br>
This is a system of Diophantine equations: 3 variables but only 2 equations; but we can find the solution(s) using the fact that the variables have to have non-negative integer values.  And in this problem, since it says that all three types of pads were sold, the values in fact must be positive integers.<br>
(1) Eliminate one of the variables to get a system of one equation in two variables;
(2) solve that equation for one variable in terms of the other; and
(3) use the fact that the variable values must be positive integers to find the solution(s).<br>
We can double the second equation and subtract the two equations to eliminate z:<br>
{{{x+y+z = 90}}}
{{{18x+4y+z = 180}}}
{{{17x+3y = 90}}}<br>
We can see by inspection that (0,30) would be a solution if x could be 0; but we know x has to be a positive integer.<br>
With the coefficients 17 and 3 being relatively prime, all other potential solutions can be found starting with (0,30) and adding 3 to the value of x while subtracting 17 from the value of y.  We get
(0,30), (3,13), (6,-4)...
But the variable values have to be positive integers.<br>
So the only solution to this problem is x=3 and y=13, which makes z=74.<br>
CHECK: 
3+13+74 = 90  check
3(9)+13(2)+74(.5) = 27+26+37 = 90  check