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Let's analyze the problem and answer each part.

### Part 1: Understanding the Solution
The system of equations given is:
\[
\begin{cases}
x + y = 4.25 \\
16.90x + 4y = 36.35
\end{cases}
\]
where:
- $ x $ represents the yards of silk fabric purchased.
- $ y $ represents the yards of cotton fabric purchased.

This system represents the constraints for the dressmaker’s purchase:
1. The first equation, $ x + y = 4.25 $, shows the total yards of fabric bought (silk and cotton combined).
2. The second equation, $ 16.90x + 4y = 36.35 $, represents the total cost, where silk costs $16.90 per yard and cotton costs $4 per yard.

**Answer to Part 1**: The solution to this system represents the specific quantities of silk ($ x $) and cotton ($ y $) fabric (in yards) that the dressmaker bought to meet both the yardage and total cost constraints.

### Part 2: Solving the System
To solve for $ x $ and $ y $, we can use substitution or elimination.

1. From the first equation, solve for $ y $:
   \[
   y = 4.25 - x
   \]

2. Substitute $ y = 4.25 - x $ into the second equation:
   \[
   16.90x + 4(4.25 - x) = 36.35
   \]

3. Simplify and solve for $ x $:
   \[
   16.90x + 17 - 4x = 36.35
   \]
   \[
   12.90x = 19.35
   \]
   \[
   x = \frac{19.35}{12.90} \approx 1.5
   \]

4. Substitute $ x = 1.5 $ back into $ y = 4.25 - x $:
   \[
   y = 4.25 - 1.5 = 2.75
   \]

So, the solution is:
\[
x = 1.5 \quad \text{and} \quad y = 2.75
\]

**Answer to Part 2**: The dressmaker bought approximately 1.5 yards of silk and 2.75 yards of cotton.

Would you like more clarification on any of these steps?

---

### Related Questions
1. How can systems of equations be solved using matrix methods?
2. What are real-life examples where systems of linear equations are used to solve budget constraints?
3. How does substitution work in solving systems, and when is it most effective?
4. What is the role of unit pricing in setting up equations like this one?
5. How can graphing help in visualizing solutions to systems of equations?

**Tip**: Always double-check the units (yards, dollars, etc.) in word problems to ensure your solution matches the context.

At a fabric store, fabrics are sold by the yard. A dressmaker spent $36.35 on 4.25 yards of silk and cotton fabrics for a dress. Silk is $16.90 per yard and cotton is $4 per yard. Here is a system of equations that represent the constraints in the situation:

\[
\begin{cases}
x + y = 4.25 \\
16.90x + 4y = 36.35
\end{cases}
\]
What does the solution to the system represent? Find the solution to the system of equations.

Learn how to solve a system of equations representing the cost and yardage of silk and cotton fabrics purchased by a dressmaker. The problem covers algebraic methods, including substitution and elimination, and applies real-world pricing. Suitable for students in Grades 8-10.

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