Answer:
 Kyle filled 4 10-oz cups, 6 14-oz cups, and 4 20-oz cups. 
Step-by-step explanation:
Let 10-oz, 14-oz, and 20-oz coffees be represented by the variables <em>a, b</em>, and <em>c</em>, respectively. 
Since a total of 14 cups of coffee was served: 

A total of 204 ounces of coffee was served. Therefore: 

A total of $16.70 was collected. Hence: 

This yields a triple system of equations. In order to solve a triple system, we should isolate the system to only two variables first. 
From the first equation, let's subtract <em>a</em> and <em>b</em> from both sides: 

Substitute this into both the second and third equations:

And: 

In this way, we've successfully created a system of two equations, which can be more easily solved. Distribute: 
For the Second Equation:
 
And for the Third: 

We can solve this using substitution. From the second equation, isolate <em>a: </em>
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Substitute into the third: 

Distribute and simplify: 

Therefore: 

Using the equation for <em>a: </em>
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And using the equation for <em>c: </em>
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Therefore, Kyle filled 4 10-oz cups, 6 14-oz cups, and 4 20-oz cups.