Determine the values of $$ x $$ and $$ y $$, in the equation: $$ \frac{x-1}{3+i}+\frac{y-1}{3-i}=i $$. Solve the following equations: i) $$ 2^{2 x+1}=3^{2 x-5} \quad $$ ii) $$ \log _{2} x-\log _{2} y=2 $$ $$ \log _{2}(x-2 y)=3 $$ In a senior secondary school, 80 students play hockey or football. The number that plays football is 5 more than twice the number that play hockey. If 15 students play both games and every student in the schoo plays at least one game, find the number of students that play football.

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UpStudy AI · Accepted Answer

Let's address each of your problems step-by-step.

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### **1. Solving for $$ x $$ and $$ y $$ in the Equation:**
$$
\frac{x-1}{3+i} + \frac{y-1}{3-i} = i
$$

**Step 1: Rationalize the Denominators**

Multiply both sides by the product of the denominators to eliminate the complex numbers in the denominators:
$$
(3+i)(3-i) = 9 - i^2 = 9 + 1 = 10
$$

Multiply both sides by 10:
$$
(x - 1)(3 - i) + (y - 1)(3 + i) = 10i
$$

**Step 2: Expand and Separate Real and Imaginary Parts**
$$
3(x - 1) - (x - 1)i + 3(y - 1) + (y - 1)i = 10i
$$
$$
[3x - 3 + 3y - 3] + [-x + 1 + y - 1]i = 10i
$$
$$
3x + 3y - 6 + (-x + y)i = 10i
$$

**Step 3: Equate Real and Imaginary Parts**
- **Real Part:**
  $$
  3x + 3y - 6 = 0 \quad \Rightarrow \quad 3x + 3y = 6 \quad \Rightarrow \quad x + y = 2
  $$
  
- **Imaginary Part:**
  $$
  -x + y = 10
  $$

**Step 4: Solve the System of Equations**
$$
\begin{cases}
x + y = 2 \
-x + y = 10
\end{cases}
$$

Add both equations:
$$
2y = 12 \quad \Rightarrow \quad y = 6
$$

Substitute $$ y = 6 $$ into the first equation:
$$
x + 6 = 2 \quad \Rightarrow \quad x = -4
$$

**Final Answer:**
$$
x = -4 \quad \text{and} \quad y = 6
$$

---

### **2. Solving the Exponential and Logarithmic Equations**

#### **i) Solving $$ 2^{2x + 1} = 3^{2x - 5} $$**
  
**Step 1: Take the Natural Logarithm of Both Sides**
$$
(2x + 1) \ln 2 = (2x - 5) \ln 3
$$

**Step 2: Rearrange Terms to Isolate $$ x $$**
$$
2x \ln 2 + \ln 2 = 2x \ln 3 - 5 \ln 3
$$
$$
2x (\ln 2 - \ln 3) = -5 \ln 3 - \ln 2
$$

**Step 3: Solve for $$ x $$**
$$
x = \frac{-5 \ln 3 - \ln 2}{2 (\ln 2 - \ln 3)} = \frac{5 \ln 3 + \ln 2}{2 (\ln 3 - \ln 2)}
$$

**Final Answer:**
$$
x = \frac{5 \ln 3 + \ln 2}{2 (\ln 3 - \ln 2)}
$$

---

#### **ii) Solving the System:**
$$
\begin{cases}
\log_2 x - \log_2 y = 2 \
\log_2 (x - 2y) = 3
\end{cases}
$$

**Step 1: Simplify the First Equation Using Logarithm Properties**
$$
\log_2 \left(\frac{x}{y}\right) = 2 \quad \Rightarrow \quad \frac{x}{y} = 2^2 = 4 \quad \Rightarrow \quad x = 4y
$$

**Step 2: Substitute $$ x = 4y $$ into the Second Equation**
$$
\log_2 (4y - 2y) = \log_2 (2y) = 3 \quad \Rightarrow \quad 2y = 2^3 = 8 \quad \Rightarrow \quad y = 4
$$

**Step 3: Find $$ x $$ Using $$ x = 4y $$**
$$
x = 4 \times 4 = 16
$$

**Final Answer:**
$$
x = 16 \quad \text{and} \quad y = 4
$$

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### **3. Senior Secondary School Sports Problem**

**Given:**
- Total students playing at least one sport: 80
- Number playing both sports: 15
- Number playing football ($$ F $$) is 5 more than twice the number playing hockey ($$ H $$): $$ F = 2H + 5 $$
  
**Using the Principle of Inclusion-Exclusion:**
$$
H + F - \text{Both} = \text{Total}
$$
$$
H + F - 15 = 80
$$
$$
H + F = 95
$$

**Substitute $$ F = 2H + 5 $$ into the Equation:**
$$
H + (2H + 5) = 95
$$
$$
3H + 5 = 95
$$
$$
3H = 90 \quad \Rightarrow \quad H = 30
$$

**Find $$ F $$:**
$$
F = 2(30) + 5 = 60 + 5 = 65
$$

**Final Answer:**
$$
\text{Number of students who play football} = 65
$$

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### **Summary of Answers**

1. **For the complex equation:**
   $$
   x = -4 \quad \text{and} \quad y = 6
   $$

2. **Solutions to the equations:**
   - **i)** $$ x = \dfrac{5 \ln 3 + \ln 2}{2 (\ln 3 - \ln 2)} $$
   - **ii)** $$ x = 16 $$ and $$ y = 4 $$

3. **Number of students who play football:**
   $$
   65 \text{ students}
   $$
**Solutions:** 1. **Complex Equation:** $$ x = -4 \quad \text{and} \quad y = 6 $$ 2. **Exponential and Logarithmic Equations:** - **i)** $$ x = \dfrac{5 \ln 3 + \ln 2}{2 (\ln 3 - \ln 2)} $$ - **ii)** $$ x = 16 $$ and $$ y = 4 $$ 3. **Sports Problem:** $$ 65 \text{ students play football} $$

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