A dietitian is planning a meal of beef, corn, and peas for a patient. The patient needs the meal to contain 95.4 grams of carbohydrate, 48.6 grams of protein and 23.2 grams of fat. The table below shows the number of grams of carbohydrate, protein, and fat in 100 grams of each food. (Note, the total is not 100 grams due to the water and fiber content of the food.) How much of each food should be used so that the patient gets the needed nutrients?
beef	 	
 g
corn	 	
 g
peas	 	
 g
Carbohydrate	Protein	Fat
Beef	1	22	12
Corn	21	3	1
Peas	12	3	1


To solve this, we need to create a system of linear equations based on the information provided. The amounts of beef, corn, and peas will be our unknowns, which we'll denote as \( x \) (grams of beef), \( y \) (grams of corn), and \( z \) (grams of peas). These will represent how much of each food should be used to meet the dietary requirements of carbohydrates, protein, and fat.

### Given Information:
1. Beef: 1 g of carbohydrate, 22 g of protein, 12 g of fat per 100 g.
2. Corn: 21 g of carbohydrate, 3 g of protein, 1 g of fat per 100 g.
3. Peas: 12 g of carbohydrate, 3 g of protein, 1 g of fat per 100 g.

The patient needs:
- 95.4 g of carbohydrates,
- 48.6 g of protein, and
- 23.2 g of fat.

### Step 1: Set up the system of equations
We can write one equation for each nutrient (carbohydrates, protein, and fat), using the proportions of each nutrient in each food.

#### Carbohydrates:
\[
\frac{1}{100}x + \frac{21}{100}y + \frac{12}{100}z = 95.4
\]
\[
0.01x + 0.21y + 0.12z = 95.4
\]

#### Protein:
\[
\frac{22}{100}x + \frac{3}{100}y + \frac{3}{100}z = 48.6
\]
\[
0.22x + 0.03y + 0.03z = 48.6
\]

#### Fat:
\[
\frac{12}{100}x + \frac{1}{100}y + \frac{1}{100}z = 23.2
\]
\[
0.12x + 0.01y + 0.01z = 23.2
\]

### Step 2: Solve the system of equations
We now have the following system of equations:
1. \( 0.01x + 0.21y + 0.12z = 95.4 \)
2. \( 0.22x + 0.03y + 0.03z = 48.6 \)
3. \( 0.12x + 0.01y + 0.01z = 23.2 \)

I'll solve this system of equations to find the values of \( x \), \( y \), and \( z \).The solution to the system of equations is:

- \( x = 150 \) grams of beef,
- \( y = 350 \) grams of corn, and
- \( z = 170 \) grams of peas.

These amounts will provide the required 95.4 grams of carbohydrates, 48.6 grams of protein, and 23.2 grams of fat.

Would you like further details, or do you have any other questions?

Here are 5 related questions for further practice:
1. How would the solution change if the patient needed 100 grams of protein?
2. What if the fat content requirement were reduced to 20 grams?
3. How much of each food is required if the patient needs only 60 grams of carbohydrates?
4. How do you verify that the given solution meets the required nutrient values?
5. How would the equations change if a fourth food option were added?

**Tip:** When solving systems of linear equations, organizing your equations carefully helps ensure accurate solutions.

A dietitian is planning a meal of beef, corn, and peas for a patient. The patient needs the meal to contain 95.4 grams of carbohydrate, 48.6 grams of protein, and 23.2 grams of fat. The table below shows the number of grams of carbohydrate, protein, and fat in 100 grams of each food. How much of each food should be used so that the patient gets the needed nutrients?
Carbohydrate    Protein    Fat
Beef     1g    22g    12g
Corn     21g   3g     1g
Peas     12g   3g     1g

Learn how to solve a system of linear equations to plan a meal containing 95.4g of carbohydrate, 48.6g of protein, and 23.2g of fat using beef, corn, and peas. Detailed step-by-step solution using algebra and systems of equations. Suitable for Grades 10-12.

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