Let's analyze and solve the two questions given in the image:

The relationship between Kelvin ($x$ in K) and Fahrenheit ($y^\circ F$) is provided by the linear equation: $y = \frac{9}{5}(x - 273) + 32$

a. Find the temperature of the liquid in Fahrenheit if the temperature of the body is 313 K.

To solve this, substitute $x = 313 \, K$ into the equation: $y = \frac{9}{5}(313 - 273) + 32 = \frac{9}{5}(40) + 32 = 72 + 32 = 104^\circ F$ So, the temperature in Fahrenheit is 104°F.

b. If the temperature is 158°F, then find the temperature in Kelvin.

Now, we reverse the process. We are given $y = 158^\circ F$ and need to find $x$ (the temperature in Kelvin): $158 = \frac{9}{5}(x - 273) + 32$ First, subtract 32 from both sides: $158 - 32 = \frac{9}{5}(x - 273)$ $126 = \frac{9}{5}(x - 273)$ Multiply both sides by $\frac{5}{9}$: $\frac{5}{9} \times 126 = x - 273$ $70 = x - 273$ Now, add 273 to both sides: $x = 70 + 273 = 343 \, K$ So, the temperature in Kelvin is 343 K.

Would you like further details or have any questions? Here are five related questions to expand on this:

1. How can you convert from Fahrenheit to Celsius using a similar formula?
2. What is the equivalent temperature of 0 Kelvin in Fahrenheit?
3. How does this conversion formula change for Celsius to Kelvin?
4. What happens when negative temperatures in Fahrenheit are converted to Kelvin?
5. Can you graph this linear relationship between Kelvin and Fahrenheit for different temperatures?

Tip: When working with temperature conversions, always ensure you use the correct constants for each formula!