A glass of skim milk supplies 0.1 mg of iron and 8.3 g of protein. A quarter pound of lean meat provides 3.2 mg of iron and 21 g of protein. A person on a special diet is to have 3.8 mg of iron and 70.8 g of protein. How many glasses of skim milk and how many​ quarter-pound servings of meat will provide​ this?


Let:
- \( x \) be the number of glasses of skim milk.
- \( y \) be the number of quarter-pound servings of lean meat.

Each glass of skim milk provides 0.1 mg of iron and 8.3 g of protein, while each quarter-pound serving of lean meat provides 3.2 mg of iron and 21 g of protein. We are tasked with finding how many glasses of milk (\( x \)) and servings of meat (\( y \)) will give a total of 3.8 mg of iron and 70.8 g of protein.

We can set up the following system of equations based on this information:

1. **Iron equation**: \( 0.1x + 3.2y = 3.8 \)
2. **Protein equation**: \( 8.3x + 21y = 70.8 \)

Now let's solve this system of equations to find \( x \) and \( y \).

### Solving the system:

**Step 1: Solve the first equation for \( x \):**

\[
0.1x + 3.2y = 3.8 \implies x = \frac{3.8 - 3.2y}{0.1}
\]

**Step 2: Substitute this expression for \( x \) into the second equation:**

\[
8.3\left(\frac{3.8 - 3.2y}{0.1}\right) + 21y = 70.8
\]

Let's simplify this equation.The solution to the system of equations is:
- \( x = 6 \) glasses of skim milk,
- \( y = 1 \) quarter-pound serving of lean meat.

This means the person should consume 6 glasses of skim milk and 1 quarter-pound serving of lean meat to meet their dietary requirement of 3.8 mg of iron and 70.8 g of protein.

Would you like further details or have any questions? Here are 5 related questions for deeper understanding:
1. How can we adjust the amounts if the protein requirement changes?
2. What if the person needed more iron? How would the equation change?
3. Can we have fractional values for servings, and how would that affect the solution?
4. How would the system of equations change with an additional food source?
5. How does the protein to iron ratio differ between milk and meat?

**Tip:** Systems of equations can be easily solved using substitution or elimination methods, especially when dealing with nutritional requirements.

A glass of skim milk supplies 0.1 mg of iron and 8.3 g of protein. A quarter pound of lean meat provides 3.2 mg of iron and 21 g of protein. A person on a special diet is to have 3.8 mg of iron and 70.8 g of protein. How many glasses of skim milk and how many​ quarter-pound servings of meat will provide​ this?

Learn how to solve a system of equations to determine how many glasses of skim milk and quarter-pound servings of lean meat meet specific iron and protein dietary requirements. This problem covers algebraic concepts like systems of equations and substitution methods, suitable for Grades 9-11.

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