Kyle works at a donut​ factory, where a​ 10-oz cup of coffee costs 95¢​, a​ 14-oz cup costs​ $1.15, and a​ 20-oz cup costs​ $1.50. During one busy​ period, Kyle served 14 cups of​ coffee, using 204 ounces of​ coffee, while collecting a total of ​$16.70. How many cups of each size did Kyle​ fill? Kyle filled ___ ​10-oz cup(s), ___ 14-oz cup(s), and ___ ​20-oz cup(s).

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 a, b, and c, 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 a and b 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 a:

Substitute into the third:

Distribute and simplify:

Therefore:

Using the equation for a:

And using the equation for c:

Therefore, Kyle filled 4 10-oz cups, 6 14-oz cups, and 4 20-oz cups.

answer: yes, since the line has a steeper rate of change than the exponential the y-values will be greater and they will only intersect one time.