A pharmacist mixes together three types of vitamin tablets. Each tablet A​ contains, among other​ things, 14 mg of niacin and 12 I.U. of vitamin E. The amounts for a tablet B are 19 mg and 15 ​I.U., and for a tablet C are 24 mg 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. 225 ​tablets, 4525 mg of​ niacin, and 5425 I.U. of vitamin E
b. 180 ​tablets, 3370 mg of​ niacin, and 3810 I.U. of vitamin E
c. 230 ​tablets, 4220 mg of​ niacin, and 4310 I.U. of vitamin E

To solve this problem, we need to set up a system of equations for each scenario (a, b, and c), using the information provided about the composition of each tablet.

### Definitions:
- Let \( x \) represent the number of **Tablet A**.
- Let \( y \) represent the number of **Tablet B**.
- Let \( z \) represent the number of **Tablet C**.

Each equation will come from the following considerations:
1. The total number of tablets.
2. The total amount of niacin.
3. The total amount of vitamin E.

### System of Equations:
For each scenario, we will have three equations based on these conditions.

#### a) 225 tablets, 4525 mg of niacin, and 5425 I.U. of vitamin E
1. **Total number of tablets**:  
   \[
   x + y + z = 225
   \]
2. **Total amount of niacin** (mg):  
   \[
   14x + 19y + 24z = 4525
   \]
3. **Total amount of vitamin E** (I.U.):  
   \[
   12x + 15y + 37z = 5425
   \]

#### b) 180 tablets, 3370 mg of niacin, and 3810 I.U. of vitamin E
1. **Total number of tablets**:  
   \[
   x + y + z = 180
   \]
2. **Total amount of niacin** (mg):  
   \[
   14x + 19y + 24z = 3370
   \]
3. **Total amount of vitamin E** (I.U.):  
   \[
   12x + 15y + 37z = 3810
   \]

#### c) 230 tablets, 4220 mg of niacin, and 4310 I.U. of vitamin E
1. **Total number of tablets**:  
   \[
   x + y + z = 230
   \]
2. **Total amount of niacin** (mg):  
   \[
   14x + 19y + 24z = 4220
   \]
3. **Total amount of vitamin E** (I.U.):  
   \[
   12x + 15y + 37z = 4310
   \]

### Solving the Systems of Equations:
For each case (a, b, and c), we can solve this system of three equations to find the values of \( x \), \( y \), and \( z \) (the number of each tablet).

Let me solve for case (a) first. I'll compute the values of \( x \), \( y \), and \( z \).It seems like I can’t do more advanced data analysis right now. Please try again later. If you want, I can guide you on how to solve the system of equations by hand or using other tools.

Would you like me to explain the steps for solving these equations manually, or do you have another question in mind?

Here are 5 related questions:
1. How can we solve a system of linear equations using substitution?
2. What is the matrix method for solving systems of equations?
3. How do we interpret the solutions of systems of equations in real-life scenarios?
4. How can you solve a system of equations using a graphing method?
5. What are the conditions under which a system of equations has no solution?

**Tip**: When dealing with large systems of equations, tools like matrix calculators or symbolic solvers (e.g., Wolfram Alpha) can quickly solve them for you!

A pharmacist mixes together three types of vitamin tablets. Each tablet A contains, among other things, 14 mg of niacin and 12 I.U. of vitamin E. The amounts for a tablet B are 19 mg and 15 I.U., and for a tablet C are 24 mg 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) 225 tablets, 4525 mg of niacin, and 5425 I.U. of vitamin E, b) 180 tablets, 3370 mg of niacin, and 3810 I.U. of vitamin E, c) 230 tablets, 4220 mg of niacin, and 4310 I.U. of vitamin E.

Learn how to solve a system of equations involving three types of vitamin tablets, given total tablets, niacin, and vitamin E amounts. This problem uses algebraic methods and is suitable for high school students in Grades 9-12.

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