https://file.aiquickdraw.com/mathbot/file/1728058006640o3dwt0cm.jpg

Let's analyze the system of equations given in the problem:

The two equations are:
\[
x + y = 4.25 \tag{1}
\]
\[
16.90x + 4y = 36.35 \tag{2}
\]

### 1. **What does the solution to the system represent?**
The variables in the system are:
- $x$: The number of yards of silk bought.
- $y$: The number of yards of cotton bought.

Equation (1) tells us that the total number of yards of fabric (both silk and cotton) is 4.25 yards. Equation (2) represents the total cost of the fabrics, where the cost per yard of silk is \$16.90, and the cost per yard of cotton is \$4. The total cost was \$36.35. 

So, the solution to the system represents the exact number of yards of silk ($x$) and the number of yards of cotton ($y$) that the dressmaker bought.

### 2. **Find the solution to the system of equations.**
To solve the system, we can use substitution or elimination. Here, substitution is a good approach because the first equation is already simple.

#### Step 1: Solve equation (1) for $y$:
\[
y = 4.25 - x \tag{3}
\]

#### Step 2: Substitute equation (3) into equation (2):
\[
16.90x + 4(4.25 - x) = 36.35
\]

Simplifying:
\[
16.90x + 17 - 4x = 36.35
\]
\[
(16.90 - 4)x + 17 = 36.35
\]
\[
12.90x + 17 = 36.35
\]

#### Step 3: Solve for $x$:
\[
12.90x = 36.35 - 17
\]
\[
12.90x = 19.35
\]
\[
x = \frac{19.35}{12.90} \approx 1.5
\]

#### Step 4: Substitute $x = 1.5$ back into equation (3) to find $y$:
\[
y = 4.25 - 1.5 = 2.75
\]

### Final Solution:
- The dressmaker bought **1.5 yards of silk** and **2.75 yards of cotton**.

Would you like further details or have any questions?

Here are some related questions you can explore:
1. How would the total cost change if the price of silk increased by 10%?
2. What would happen if the total amount of fabric purchased was increased to 6 yards?
3. How could the system of equations be solved using the elimination method?
4. What is the graphical interpretation of the solution to this system?
5. How can we check if the solution satisfies both equations?

**Tip**: Always double-check your solution by substituting the values of the variables back into the original equations to verify their accuracy!

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. What does the solution to the system represent? Find the solution to the system of equations: x + y = 4.25, 16.90x + 4y = 36.35.

Learn how to solve a system of linear equations involving the cost of silk and cotton fabrics. This problem uses algebraic concepts to determine how many yards of each fabric the dressmaker bought, with silk at $16.90 per yard and cotton at $4 per yard. Step-by-step solution provided for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation