A jeweler had a fixed amount of gold to make bracelets and necklaces. The amount of gold in each bracelet is 7 grams and the amount of gold in each necklace is 18 grams. The jeweler made a total of 7 bracelets and necklaces using 82 grams of gold. Determine the number of bracelets made and the number of necklaces made.

Let's denote:
- \( x \) as the number of bracelets made.
- \( y \) as the number of necklaces made.

We know two things:
1. The total number of bracelets and necklaces is 7:
   \[
   x + y = 7
   \]

2. The total amount of gold used is 82 grams, with each bracelet using 7 grams and each necklace using 18 grams:
   \[
   7x + 18y = 82
   \]

We now have a system of two equations:

\[
\begin{cases}
x + y = 7 \\
7x + 18y = 82
\end{cases}
\]

Let's solve this system step-by-step.

### Step 1: Solve for one variable

First, solve for \( y \) in terms of \( x \) using the first equation:

\[
y = 7 - x
\]

### Step 2: Substitute into the second equation

Substitute \( y = 7 - x \) into the second equation:

\[
7x + 18(7 - x) = 82
\]

Expand and simplify:

\[
7x + 126 - 18x = 82
\]

Combine like terms:

\[
-11x + 126 = 82
\]

### Step 3: Solve for \( x \)

Rearrange to solve for \( x \):

\[
-11x = 82 - 126
\]

\[
-11x = -44
\]

\[
x = 4
\]

### Step 4: Solve for \( y \)

Substitute \( x = 4 \) back into the equation \( y = 7 - x \):

\[
y = 7 - 4 = 3
\]

### Solution

The jeweler made:
- **4 bracelets**.
- **3 necklaces**.

### Verification

To check the solution:
- Total items: \( 4 + 3 = 7 \) (Correct)
- Total gold used: \( 7 \times 4 + 18 \times 3 = 28 + 54 = 82 \) grams (Correct)

The solution is consistent with the given conditions.

Would you like more details, or do you have any questions?

Here are 5 related questions that might interest you:
1. What if the jeweler had 100 grams of gold? How many bracelets and necklaces could be made?
2. If each necklace costs 3 times more than each bracelet, what is the cost ratio?
3. How would the solution change if the jeweler decided to make at least 5 necklaces?
4. What is the minimum amount of gold needed to make 10 items?
5. How would the solution change if each bracelet required 8 grams of gold instead of 7?

**Tip:** When solving word problems, always define your variables clearly and check your solution against the problem's conditions.

Learn how to solve a system of linear equations to determine how many bracelets and necklaces a jeweler made using 82 grams of gold. This problem involves key algebraic concepts such as substitution and linear equations. Suitable for students in Grades 7-9.

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