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Let's go through the questions one by one using the provided formula:

### Formula:
\[
C = \frac{5}{9}(F - 160)
\]
Where:
- \(C\) is the temperature in Celsius.
- \(F\) is the temperature in Fahrenheit.

### a. If the temperature is 86°F, what is the temperature in Celsius?

Using \(F = 86\) in the equation:
\[
C = \frac{5}{9}(86 - 160)
\]
Let me calculate that:

\[
C = \frac{5}{9}(-74)
\]

\[
C = -41.11^\circ C
\]

So, the temperature is approximately **\(-41.11^\circ C\)**.

---

### b. If the temperature is 35°C, what is the temperature in Fahrenheit?

To convert from Celsius to Fahrenheit, we use the inverse formula:
\[
F = \frac{9}{5}C + 32
\]
Substitute \(C = 35\):
\[
F = \frac{9}{5}(35) + 32
\]

Let me calculate:

\[
F = 63 + 32 = 95^\circ F
\]

So, the temperature is **95°F**.

---

### c. If the temperature is 0°C, what is the temperature in Fahrenheit, and if the temperature is 0°F, what is the temperature in Celsius?

1. **Converting 0°C to Fahrenheit:**

Using the same conversion formula \(F = \frac{9}{5}C + 32\):
\[
F = \frac{9}{5}(0) + 32 = 32^\circ F
\]

So, 0°C is **32°F**.

2. **Converting 0°F to Celsius:**

Using \(F = 0\) in the original formula:
\[
C = \frac{5}{9}(0 - 160) = \frac{5}{9}(-160) = -88.89^\circ C
\]

So, 0°F is approximately **\(-88.89^\circ C\)**.

---

### d. What is the numerical value of the temperature which is the same in both scales?

For temperatures where \(F = C\), set \(F = C\) in the formula \(C = \frac{5}{9}(F - 160)\). This gives:
\[
C = \frac{5}{9}(C - 160)
\]
Multiplying both sides by 9 to eliminate the denominator:
\[
9C = 5(C - 160)
\]
Expanding:
\[
9C = 5C - 800
\]
Bringing all terms involving \(C\) to one side:
\[
4C = -800
\]
Solving for \(C\):
\[
C = -200
\]

So, the temperature is **\(-200^\circ\)** where Fahrenheit and Celsius are equal.

---

Would you like more details on any part of these solutions?

### Related questions:
1. How would the formula change if converting between Kelvin and Celsius?
2. If 100°F is measured, how would you convert it to Celsius?
3. What is the boiling point of water in Fahrenheit and Celsius, and how do they compare?
4. Can we express the conversion formulas in terms of \(C\) alone, without Fahrenheit?
5. How do Celsius and Fahrenheit scales differ in how they represent freezing and boiling points of water?

### Tip:
To quickly estimate Celsius from Fahrenheit, subtract 32 from the Fahrenheit value and multiply by 5/9. For Fahrenheit from Celsius, reverse this by multiplying by 9/5 and adding 32!

a. If the temperature is 86°F, what is the temperature in Celsius?
b. If the temperature is 35°C, what is the temperature in Fahrenheit?
c. If the temperature is 0°C what is the temperature in Fahrenheit and if the temperature is 0°F, what is the temperature in Celsius?
d. What is the numerical value of the temperature which is same in both the scales?

This problem covers temperature conversions between Fahrenheit and Celsius using linear equations. Learn to solve for Celsius from Fahrenheit and vice versa, including special cases where the temperatures are equal in both scales.

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