Help me set up the matrices for this! I can solve it, but having a difficult time getting the matrix set up. A pharmacist mixes together three types of vitamin tablets. Each Tablet A contains, among other things, 12 mg of niacin and 10 I.U. of vitamin E. The amounts for a tablet B are 17mg and 13 IU and for tablet C are 22mg and 35 IU. How many of each tablet if there are total of 225 tablets with a total amount of 4075mg niacin and 4975IU?

Question

Help me set up the matrices for this! I can solve it, but having a difficult time getting the matrix set up. A pharmacist mixes together three types of vitamin tablets. Each Tablet A contains, among other things, 12 mg of niacin and 10 I.U. of vitamin E. The amounts for a tablet B are 17mg and 13 IU and for tablet C are 22mg and 35 IU. How many of each tablet if there are total of 225 tablets with a total amount of 4075mg niacin and 4975IU?

Answer 1

count the tablets and the ingredients:

a + b + c = 225
12a + 17b + 22c = 4075
10a + 13b + 35c = 4975

Now just use your matrix tools.

Answer 2

To set up the matrices for this problem, we can define variables to represent the number of tablets of each type. Let's use the following variables:

Let x represent the number of tablets of type A.
Let y represent the number of tablets of type B.
Let z represent the number of tablets of type C.

Now, let's set up the equations based on the given information.

1) Niacin equation:
The total amount of niacin in the tablets can be found by multiplying the niacin content of each tablet by the number of tablets of that type and then summing them up. So we have the equation:

12x + 17y + 22z = 4075 (since the total niacin content is 4075mg)

2) Vitamin E equation:
Similar to the niacin equation, the total amount of vitamin E can be found by multiplying the vitamin E content of each tablet by the number of tablets of that type and then summing them up. So we have the equation:

10x + 13y + 35z = 4975 (since the total vitamin E content is 4975IU)

3) Total number of tablets equation:
The total number of tablets is given as 225, so we have the equation:

x + y + z = 225

Now that we have three equations, we can set up the matrix to solve for x, y, and z using matrix methods. The matrix equation will be of the form AX = B, where A is the coefficient matrix, X is the matrix of variables, and B is the matrix of constants.

The coefficient matrix A will be:

[12 17 22 ]
[10 13 35 ]
[ 1 1 1 ]

The matrix of variables X will be:

[ x ]
[ y ]
[ z ]

The matrix of constants B will be:

[ 4075 ]
[ 4975 ]
[ 225 ]

Now, you can solve the system of equations using matrix operations such as Gaussian elimination, inverse matrix, or matrix calculator to find the values for x, y, and z.