-/2 Points] DETAILS MY NOTES KAUFACS10 2.3.050. YOUR TEACHER PRACTICE ANOTHER Use an algebraic approach to solve the problem. Abby has 51 coins, consisting only of dimes and quarters, worth $$ \$ 9.45 $$. How many dimes and how many quarters does she have? dimes quarters Additional Materials eBook

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Sure, let's solve the problem step-by-step using an algebraic approach.

**Problem Statement:**
Abby has a total of 51 coins, consisting only of dimes and quarters, worth \$9.45. We need to determine how many dimes and how many quarters she has.

**Let’s define our variables:**
- Let $$ d $$ represent the number of dimes.
- Let $$ q $$ represent the number of quarters.

**Step 1: Set Up the Equations**

We have two pieces of information which we'll translate into equations:

1. **Total number of coins:**
   $$
   d + q = 51
   $$

2. **Total value of the coins:**
   - Each dime is worth \$0.10.
   - Each quarter is worth \$0.25.
   - The total value is \$9.45.
   
   So,
   $$
   0.10d + 0.25q = 9.45
   $$

**Step 2: Solve the System of Equations**

We have the following system:
$$
\begin{cases}
d + q = 51 \quad \text{(1)} \
0.10d + 0.25q = 9.45 \quad \text{(2)}
\end{cases}
$$

Let's solve equation (1) for one of the variables, say $$ d $$:

$$
d = 51 - q
$$

Now, substitute this expression for $$ d $$ into equation (2):

$$
0.10(51 - q) + 0.25q = 9.45
$$

**Step 3: Simplify and Solve for $$ q $$**

Distribute the 0.10:

$$
0.10 \times 51 - 0.10q + 0.25q = 9.45
$$
$$
5.10 - 0.10q + 0.25q = 9.45
$$

Combine like terms:

$$
5.10 + 0.15q = 9.45
$$

Subtract 5.10 from both sides:

$$
0.15q = 9.45 - 5.10
$$
$$
0.15q = 4.35
$$

Now, divide both sides by 0.15 to solve for $$ q $$:

$$
q = \frac{4.35}{0.15}
$$
$$
q = 29
$$

**Step 4: Find $$ d $$ Using the Value of $$ q $$**

Now that we know $$ q = 29 $$, substitute this back into the expression for $$ d $$:

$$
d = 51 - q
$$
$$
d = 51 - 29
$$
$$
d = 22
$$

**Solution:**
- **Number of dimes ($$ d $$) = 22**
- **Number of quarters ($$ q $$) = 29**

**Verification:**

Let's verify the total value to ensure our solution is correct.

- **Value from dimes:**
  $$
  22 \text{ dimes} \times \$0.10 = \$2.20
  $$

- **Value from quarters:**
  $$
  29 \text{ quarters} \times \$0.25 = \$7.25
  $$

- **Total value:**
  $$
  \$2.20 + \$7.25 = \$9.45
  $$

The total value matches the given \$9.45, so our solution is correct.

**Final Answer:**
Abby has **22 dimes** and **29 quarters**.
Abby has 22 dimes and 29 quarters.

-/2 Points] DETAILS MY NOTES KAUFACS10 2.3.050. YOUR TEACHER PRACTICE ANOTHER Use an algebraic approach to solve the problem. Abby has 51 coins, consisting only of dimes and quarters, worth \( \$ 9.45 \). How many dimes and how many quarters does she have? dimes quarters Additional Materials eBook

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