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Problem 1

# You manage an ice cream factory that makes three flavors: Creamy Vanilla, Continental Mocha, and Succulent Strawberry. Into each batch of Creamy Vanilla go eggs, 1 cup of milk, and 2 cups of cream. Into each batch of Continental Mocha go 1 egg, 1 cup of milk, and 2 cups of cream, while into each batch of Succulent Strawberry go egg, 2 cups of milk, and 1 cup of cream. You have in stock 350 eggs, 350 cups of milk, and 400 cups of cream. How many batches of each flavor should you make in order to use up all of your ingredients? HIIT [See Example 1.]

Expert verified
It is not possible to fully use up all ingredients by making batches of these ice cream flavors. There is no combination of Creamy Vanilla, Continental Mocha, and Succulent Strawberry batches that would require exactly 350 eggs, 350 cups of milk, and 400 cups of cream.
See the step by step solution

## Step 1: Define the variables

Let's assign variables to represent the number of batches of each ice cream flavor: - x: Creamy Vanilla - y: Continental Mocha - z: Succulent Strawberry

## Step 2: Write down the equations based on the given information

We can create three equations based on the provided information. Since we want to use up all ingredients, we can create an equation for each ingredient: - Eggs: $$1x + 1y + 1z = 350$$ - Milk: $$1x + 1y + 2z = 350$$ - Cream: $$2x + 2y + 1z = 400$$

## Step 3: Solve the system of equations

Let's solve the system of equations using the method of substitution or elimination. We can first eliminate the variable z from the first and second equations. This can be done by subtracting equation (1) from equation (2). $$(1x+1y+2z)-(1x+1y+1z) = (350)-(350)$$ This simplifies to: $$z = 0$$ Now we know there are no batches of Succulent Strawberry made. We can substitute z = 0 into the first equation. $$1x + 1y + 0 = 350$$ This simplifies to: $$x + y = 350$$ And substitute z = 0 into the third equation as well: $$2x + 2y + 0 = 400$$ This simplifies to: $$x + y = 200$$ Now we have two equations with two variables, x and y: 1. $$x + y = 350$$ 2. $$x + y = 200$$ We can see that there is no solution for this system of equations as if we subtract equation (2) from equation (1), we get: $$(x+y)-(x+y) = (350)-(200)$$ $$0 = 150$$

## Step 4: Interpret the results

The system of equations has no solution, meaning that it is not possible to fully use up all ingredients by making batches of these ice cream flavors. There is no combination of Creamy Vanilla, Continental Mocha, and Succulent Strawberry batches that would require exactly 350 eggs, 350 cups of milk, and 400 cups of cream.

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