Solution of System of Linear Equations

Nutrition Each ounce of substance A supplies 5% of the nutrition a patient needs. Substance B supplies 15% of the required nutrition per ounce, and substance C supplies 12% of the required nutrition per ounce. If digestive restrictions require that substances A and C be given in equal amounts, and the amount of substance B be one-fifth of either of these other amounts, find the number of ounces of each substance that should be in the meal to provide 100% of the required nutrition.

Let the number of ounces of substance A be denoted by \( x \), substance B by \( y \), and substance C by \( z \).

### Step 1: Set up the equations

We know from the problem:

1. Substance A supplies 5% of the required nutrition per ounce, so the contribution of substance A is \( 0.05x \).
2. Substance B supplies 15% of the required nutrition per ounce, so the contribution of substance B is \( 0.15y \).
3. Substance C supplies 12% of the required nutrition per ounce, so the contribution of substance C is \( 0.12z \).

The total contribution of all three substances must sum to 100%, so:

\[
0.05x + 0.15y + 0.12z = 1 \quad \text{(Equation 1: for 100% nutrition)}
\]

### Step 2: Use the given restrictions

- **First restriction**: Substances A and C must be given in equal amounts:
  \[
  x = z \quad \text{(Equation 2)}
  \]

- **Second restriction**: The amount of substance B must be one-fifth of either of the amounts of A or C:
  \[
  y = \frac{1}{5}x \quad \text{(Equation 3)}
  \]

### Step 3: Substitute the restrictions into Equation 1

Using \( x = z \) and \( y = \frac{1}{5}x \), substitute into Equation 1:

\[
0.05x + 0.15\left(\frac{1}{5}x\right) + 0.12x = 1
\]

### Step 4: Simplify the equation

Simplify the terms:

\[
0.05x + 0.15 \times \frac{1}{5}x + 0.12x = 1
\]
\[
0.05x + 0.03x + 0.12x = 1
\]
\[
0.20x = 1
\]

### Step 5: Solve for \( x \)

\[
x = \frac{1}{0.20} = 5
\]

### Step 6: Find \( y \) and \( z \)

From \( x = z \), we know that:
\[
z = 5
\]

From \( y = \frac{1}{5}x \), we know that:
\[
y = \frac{1}{5} \times 5 = 1
\]

### Final Answer

The number of ounces of each substance is:
- Substance A: \( 5 \) ounces
- Substance B: \( 1 \) ounce
- Substance C: \( 5 \) ounces

Would you like further details or have any questions? Here are some related questions:

1. What happens if the nutrition requirements change for one of the substances?
2. How would the system change if substances A and C didn’t have to be equal?
3. How can you express the problem using matrices?
4. What if substance B had a different percentage contribution?
5. How can you verify the solution using substitution?

**Tip**: Always check that all constraints are satisfied when solving a system of linear equations!

Nutrition Each ounce of substance A supplies 5% of the nutrition a patient needs. Substance B supplies 15% of the required nutrition per ounce, and substance C supplies 12% of the required nutrition per ounce. If digestive restrictions require that substances A and C be given in equal amounts, and the amount of substance B be one-fifth of either of these other amounts, find the number of ounces of each substance that should be in the meal to provide 100% of the required nutrition.

Solve a system of linear equations to determine how many ounces of substances A, B, and C are needed to provide 100% of a patient's nutritional requirements. This problem involves solving equations using substitution and is suitable for students in Grades 9-12.

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