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The document you uploaded, *MAT 205 Lab Assignment #3*, involves solving problems related to linear combinations, span, and linear independence, particularly using concrete mix examples. Here is a breakdown of what you will need to solve:

### Summary of Questions:
1. **Custom Concrete Mix**  
   - You are given a customer request for specific amounts of cement, water, sand, gravel, and fly ash, and you need to calculate the proportions of the three basic concrete mixes (Type S, Type A, Type L) that combine to meet the requested composition. You are asked to verify if the solution is unique.
   
2. **Feasibility of Various Custom Mixes**  
   - For four different custom mixes, you need to determine whether they can be made using combinations of the three basic mixes and explain your reasoning.

3. **Necessity of Three Basic Mixes**  
   - Investigate whether all three basic mixes are required by considering linear independence.

4. **Combinatory Limitations**  
   - Determine whether the three basic mixes can create all possible mixes by considering their span.

5. **Proportional Composition of Water, Sand, Gravel, Cement, and Fly Ash**  
   - Explain why the given proportionality of the materials (water, sand + gravel, cement + fly ash) holds true for any custom mix.

6. **Special Case Custom Mix**  
   - Examine if there exists a custom mix that meets the proportional conditions in question 5 but cannot be formed from the three basic mixes.

### Solution Approach:
1. **Set up a system of linear equations** for each component (cement, water, sand, gravel, and fly ash). You will need to solve for the number of scoops of Type S, Type A, and Type L that satisfy the custom mix requirement.
   
2. For **checking if mixes can be made**, you will set up another system of equations for each custom mix and solve for the consistency of the system.

3. **Linear independence** can be assessed by checking if any of the basic mixes can be written as a combination of the others.

4. **Span** of the basic mixes will show whether they cover the entire space of possible concrete mixes.

Would you like help solving a specific question or further details on how to approach them?

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### 5 Related Questions:
1. What steps are needed to solve the system of linear equations for the first question?
2. How do you interpret the uniqueness of a solution in the context of linear combinations?
3. How do you check the span of a set of vectors in linear algebra?
4. Can you give an example of a real-world application of linear independence outside concrete mixes?
5. What is the significance of the proportions given in question 5 (e.g., one-sixth water)?

### Tip:
When solving a system of linear equations, Gaussian elimination is a common method used to simplify and find solutions efficiently.

Suppose that a customer requests a custom mix containing the following amounts of cement, water, sand, gravel, and fly ash per 60 grams of mix: 16g, 10g, 21g, 9g, and 4g. Find the amounts (in grams) of Type S, Type A, and Type L that would need to be mixed together to create each 60 gram scoop of custom mix containing 16g cement, 10g water, 21g sand, 9g gravel, and 4g fly ash. Is the answer unique?

Learn how to calculate the proportions of basic concrete mixes (Type S, Type A, and Type L) to create a custom concrete mix with 16g cement, 10g water, 21g sand, 9g gravel, and 4g fly ash per 60g scoop. This problem covers concepts of linear combinations, span, and linear independence using matrix representation and Gaussian elimination.

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