he equation $$ 15-y=8 $$. Explain how you can solve the equation using the nd Subtraction Properties of Equality. Add y to each Side the addition foperty of equanity thensuptract 8 eatch side using the subtraction property of 26. DIG DEEPER An amusement park gives everyone who rides a new roller coaster at full price a $$ 20 \% $$ discount on all other rides. The prices of 4 rides at the park before the discount are shown. You ride the roller coaster once and then 3 other rides. You spend $$ \$ 10 $$. Identify several possible prices to ride the roller coaster. Which do you

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UpStudy AI · Accepted Answer

Certainly! Let's address both parts of your query step by step.

---

## 1. Solving the Equation $$15 - y = 8$$ Using Properties of Equality

### **Understanding the Equation**
You're given the equation:
$$ 15 - y = 8 $$
And you're asked to solve for $$ y $$ using the **Addition and Subtraction Properties of Equality**.

### **Step-by-Step Solution**

**Step 1: Subtract 15 from Both Sides**
To isolate $$ y $$, start by eliminating the constant term on the left side.

$$
\begin{align*}
15 - y &= 8 \
15 - 15 - y &= 8 - 15 \quad &\text{(Subtract 15 from both sides)} \
- y &= -7
\end{align*}
$$

**Step 2: Multiply Both Sides by -1**
Now, solve for $$ y $$ by getting rid of the negative sign.

$$
\begin{align*}
- y &= -7 \
(-1)(- y) &= (-1)(-7) \quad &\text{(Multiply both sides by -1)} \
y &= 7
\end{align*}
$$

**Alternative Method Using Addition:**

**Step 1: Add $$ y $$ to Both Sides**
Instead of subtracting 15 first, you can rearrange the equation by adding $$ y $$ to both sides.

$$
\begin{align*}
15 - y + y &= 8 + y \quad &\text{(Add $$ y $$ to both sides)} \
15 &= y + 8
\end{align*}
$$

**Step 2: Subtract 8 from Both Sides**
Now, isolate $$ y $$:

$$
\begin{align*}
15 - 8 &= y + 8 - 8 \quad &\text{(Subtract 8 from both sides)} \
7 &= y
\end{align*}
$$

**Conclusion:**
Regardless of the method, the solution is:
$$
y = 7
$$

---

## 2. Amusement Park Pricing Problem

### **Problem Statement:**
An amusement park offers a new roller coaster ride at full price. Everyone who rides the roller coaster receives a **20% discount** on all other rides. There are 4 other rides at the park with known prices. You ride the roller coaster once and 3 other rides, spending a total of **\$10**.

**Objective:** Identify several possible prices for the roller coaster ride.

### **Assumptions:**
1. Let $$ C $$ be the price of the roller coaster (full price).
2. Let $$ p_1, p_2, p_3, p_4 $$ be the original prices of the other 4 rides.
3. When you ride the roller coaster, you get a 20% discount on any of the other rides you choose.

### **Calculations:**

Since you rode the roller coaster once and 3 other rides with a 20% discount, your total spending can be expressed as:

$$
C + 0.8p_a + 0.8p_b + 0.8p_c = 10
$$

Where $$ p_a, p_b, p_c $$ are the prices of the three discounted rides you selected.

**Without specific prices for the other rides**, we'll explore possible scenarios by making some assumptions.

### **Scenario 1: All Other Rides Have the Same Price**

Assume all other rides are priced equally at $$ p $$.

$$
C + 3(0.8p) = 10 \
C + 2.4p = 10
$$

**Possible Combinations:**

1. **If $$ p = \$2 $$:**
   $$
   C + 2.4 \times 2 = 10 \
   C + 4.8 = 10 \
   C = 5.2
   $$
   *Roller Coaster Price: \$5.20*

2. **If $$ p = \$1 $$:**
   $$
   C + 2.4 \times 1 = 10 \
   C + 2.4 = 10 \
   C = 7.6
   $$
   *Roller Coaster Price: \$7.60*

3. **If $$ p = \$3 $$:**
   $$
   C + 2.4 \times 3 = 10 \
   C + 7.2 = 10 \
   C = 2.8
   $$
   *Roller Coaster Price: \$2.80*

### **Scenario 2: Different Prices for Other Rides**

Suppose the three discounted rides you choose have different original prices. For example:

- Ride 1: \$1
- Ride 2: \$2
- Ride 3: \$3

$$
C + 0.8(1) + 0.8(2) + 0.8(3) = 10 \
C + 0.8 + 1.6 + 2.4 = 10 \
C + 4.8 = 10 \
C = 5.2
$$
*Roller Coaster Price: \$5.20*

### **Scenario 3: Varying Discounted Ride Prices**

If the discounted rides have varying prices, the roller coaster price can adjust accordingly. For example:

- Ride 1: \$1.50
- Ride 2: \$2.50
- Ride 3: \$3.50

$$
C + 0.8(1.5) + 0.8(2.5) + 0.8(3.5) = 10 \
C + 1.2 + 2.0 + 2.8 = 10 \
C + 6.0 = 10 \
C = 4.0
$$
*Roller Coaster Price: \$4.00*

### **Conclusion:**

Given the total spending of \$10 and knowing that you receive a 20% discount on three other rides, the **price of the roller coaster ($$ C $$) can vary** depending on the original prices of the other rides you've chosen. Here are some possible prices for the roller coaster based on different scenarios:

- **\$2.80**
- **\$4.00**
- **\$5.20**
- **\$7.60**

These examples illustrate how varying the prices of the discounted rides affects the roller coaster's price. To determine the exact price, more information about the original prices of the other rides would be necessary.

---

I hope this clarifies both solving the equation using properties of equality and understanding the amusement park pricing scenario! If you have any further questions or need additional assistance, feel free to ask.
To solve $$15 - y = 8$$, subtract 15 from both sides to get $$-y = -7$$, then multiply both sides by -1 to find $$y = 7$$. For the amusement park problem, if you rode the roller coaster once and three other rides with a 20% discount, and spent a total of \$10, the roller coaster's price can be \$2.80, \$4.00, \$5.20, or \$7.60 depending on the prices of the other rides.

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