7. Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. Karen wants to buy a charm bracelet. Dazzling Diamonds charges $$ \$ 17 $$ per charm, plus $$ \$ 54 $$ for the bracelet. Jared's Jewelry charges $$ \$ 16 $$ per charm and $$ \$ 65 $$ for the bracelet. If Karen wants to add a certain number of charms to her bracelet, the cost will be the same at either Jewelry shop. How many charms would that be? What would the total cost of the bracelet be? If Karen adds

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UpStudy AI · Accepted Answer

Sure, let's break down the problem step by step.

### **1. Define the Variables**
- Let $$ c $$ represent the number of charms Karen wants to add.
- Let $$ C $$ represent the total cost of the bracelet.

### **2. Write the System of Equations**
Based on the pricing from both jewelry shops:

- **Dazzling Diamonds:**
  $$
  C = 17c + 54
  $$
  - **\$17 per charm**
  - **\$54 for the bracelet**

- **Jared's Jewelry:**
  $$
  C = 16c + 65
  $$
  - **\$16 per charm**
  - **\$65 for the bracelet**

Since Karen wants the total costs to be the same at both shops, we can set the two equations equal to each other:

$$
17c + 54 = 16c + 65
$$

### **3. Solve the System Using Substitution**

Let's solve for $$ c $$:

1. **Subtract $$ 16c $$ from both sides:**
   $$
   17c - 16c + 54 = 65
   $$
   $$
   c + 54 = 65
   $$

2. **Subtract 54 from both sides:**
   $$
   c = 65 - 54
   $$
   $$
   c = 11
   $$

### **4. Determine the Total Cost**

Now that we know $$ c = 11 $$ charms, plug this value back into either of the original equations to find the total cost $$ C $$.

Using **Dazzling Diamonds**:
$$
C = 17(11) + 54
$$
$$
C = 187 + 54
$$
$$
C = \$241
$$

*Alternatively, using **Jared's Jewelry**:*
$$
C = 16(11) + 65
$$
$$
C = 176 + 65
$$
$$
C = \$241
$$

Both calculations confirm that the total cost is **\$241**.

### **5. Summary of Answers**
- **Number of Charms:** 11
- **Total Cost of the Bracelet:** \$241
Karen needs to add **11 charms** to her bracelet, and the total cost will be **\$241** at either jewelry shop.

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