You manage an ice cream factory that makes two flavors: Creamy Vanilla and Continental Mocha. Into each quart of Creamy Vanilla go 2 eggs and 3 cups of cream. Into each quart of Continental Mocha go 1 egg and 3 cups of cream. You have in stock 600 eggs and 1,050 cups of cream. How many quarts of each flavor should you make in order to use up all the eggs and cream?
Creamy Vanilla
qt
Continental Mocha
qt

To solve this, let's set up the problem as a system of linear equations.

### Step 1: Define the Variables
Let:
- \( x \) = the number of quarts of Creamy Vanilla
- \( y \) = the number of quarts of Continental Mocha

### Step 2: Set Up the Equations Based on Ingredients
Each quart of Creamy Vanilla requires:
- 2 eggs and 3 cups of cream

Each quart of Continental Mocha requires:
- 1 egg and 3 cups of cream

You have:
- 600 eggs
- 1,050 cups of cream

The equations based on these resources are:
1. **Eggs equation:** \( 2x + y = 600 \)
2. **Cream equation:** \( 3x + 3y = 1050 \)

### Step 3: Simplify the Equations
The cream equation can be simplified by dividing everything by 3:
\[
x + y = 350
\]

So, the system of equations is:
1. \( 2x + y = 600 \)
2. \( x + y = 350 \)

### Step 4: Solve the System of Equations
From the second equation, solve for \( y \):
\[
y = 350 - x
\]

Substitute \( y = 350 - x \) into the first equation:
\[
2x + (350 - x) = 600
\]
\[
2x - x = 600 - 350
\]
\[
x = 250
\]

Now, substitute \( x = 250 \) back into the equation \( y = 350 - x \):
\[
y = 350 - 250 = 100
\]

### Solution
- **Creamy Vanilla:** \( x = 250 \) quarts
- **Continental Mocha:** \( y = 100 \) quarts

You should make 250 quarts of Creamy Vanilla and 100 quarts of Continental Mocha to use up all the eggs and cream.

Would you like further details or have any questions?

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### Related Questions
1. How many eggs would you need if you wanted to make 300 quarts of Creamy Vanilla?
2. What would happen if you had only 500 eggs in stock?
3. How many cups of cream are needed per 100 quarts of each flavor?
4. If you increased the cream requirement for Continental Mocha, how would the equations change?
5. What if you had only 1,000 cups of cream? How would the quantities change?

### Tip
When solving systems of equations, substitution or elimination can simplify finding solutions, especially when one variable can be isolated easily.

You manage an ice cream factory that makes two flavors: Creamy Vanilla and Continental Mocha. Into each quart of Creamy Vanilla go 2 eggs and 3 cups of cream. Into each quart of Continental Mocha go 1 egg and 3 cups of cream. You have in stock 600 eggs and 1,050 cups of cream. How many quarts of each flavor should you make in order to use up all the eggs and cream?

Determine how many quarts of Creamy Vanilla and Continental Mocha to produce using all 600 eggs and 1,050 cups of cream. Learn to solve a real-world problem with systems of linear equations, applying algebraic techniques and substitution. Ideal for students in Grades 8-10.

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