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Let's break this problem down step by step.

The total amounts of tablets, niacin, and vitamin E are given for three cases:

- Tablet A: 15 mg of niacin and 12 I.U. of vitamin E.
- Tablet B: 20 mg of niacin and 15 I.U. of vitamin E.
- Tablet C: 25 mg of niacin and 37 I.U. of vitamin E.

### First Set of Conditions:
- **Total tablets:** 165
- **Total niacin:** 3550 mg
- **Total vitamin E:** 4145 I.U.

### Variables:
Let:
- \( x \) be the number of tablet A
- \( y \) be the number of tablet B
- \( z \) be the number of tablet C

### System of Equations:
From the problem statement, we can set up the following system of equations:

1. **Total tablets:**
   \[
   x + y + z = 165
   \]
2. **Total niacin:**
   \[
   15x + 20y + 25z = 3550
   \]
3. **Total vitamin E:**
   \[
   12x + 15y + 37z = 4145
   \]

Now that we have this system of equations, we can solve it for \(x\), \(y\), and \(z\).

Would you like me to solve this step-by-step or provide further explanations on this approach? 

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Here are 5 questions to think about:
1. How can we verify the solution to a system of equations?
2. What methods can we use to solve this system, such as substitution or matrix methods?
3. How do the different compositions of each tablet affect the total amounts of niacin and vitamin E?
4. How might the problem change if we alter the amounts in one of the tablets?
5. Could you use a different approach, such as trial and error or estimation, to solve this system?

### Tip:
Always double-check your system of equations before solving, ensuring that each equation corresponds accurately to the conditions given in the problem.

A pharmacist mixes three types of vitamin tablets. Each tablet A contains, among other things, 15 mg of niacin and 12 I.U. of vitamin E. The amounts for a tablet B are 20 mg of niacin and 15 I.U., and for a tablet C are 25 mg of niacin and 37 I.U. How many of each tablet are there if the total number of tablets, total amount of niacin, and total amount of vitamin E are as given below:

a. 165 tablets, 3550 mg of niacin, and 4145 I.U. of vitamin E
b. 180 tablets, 3550 mg of niacin, and 3810 I.U. of vitamin E
c. 260 tablets, 5050 mg of niacin, and 4950 I.U. of vitamin E

Learn how to solve a system of equations to determine the number of vitamin A, B, and C tablets based on the total tablets, niacin, and vitamin E amounts. This problem is suitable for students in Grades 10-12 and involves solving systems of linear equations.

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