Let's begin by substituting the second equation, $V_{\text{OUT}} = V_{\text{IL}} + \frac{V_{\text{DD}}}{2}$, into the first equation and then solving for $V_{\text{IL}}$.

Step 1: First Equation

$(V_{\text{IL}} - V_{\text{T}})^2 / 2 = \left[ (V_{\text{DD}} - V_{\text{IL}} - V_{\text{T}}) (V_{\text{DD}} - V_{\text{OUT}}) - \frac{(V_{\text{DD}} - V_{\text{OUT}})^2}{2} \right]$

Step 2: Substitute $V_{\text{OUT}} = V_{\text{IL}} + \frac{V_{\text{DD}}}{2}$

Now, substitute this into the equation: $(V_{\text{IL}} - V_{\text{T}})^2 / 2 = \left[ (V_{\text{DD}} - V_{\text{IL}} - V_{\text{T}}) \left( V_{\text{DD}} - \left( V_{\text{IL}} + \frac{V_{\text{DD}}}{2} \right) \right) - \frac{\left( V_{\text{DD}} - \left( V_{\text{IL}} + \frac{V_{\text{DD}}}{2} \right) \right)^2}{2} \right]$

Step 3: Simplify $V_{\text{DD}} - V_{\text{OUT}}$

Simplifying $V_{\text{DD}} - V_{\text{OUT}}$ using $V_{\text{OUT}} = V_{\text{IL}} + \frac{V_{\text{DD}}}{2}$, we get: $V_{\text{DD}} - V_{\text{OUT}} = V_{\text{DD}} - \left( V_{\text{IL}} + \frac{V_{\text{DD}}}{2} \right) = \frac{V_{\text{DD}}}{2} - V_{\text{IL}}$

Step 4: Substitute into the Equation

Now substitute this into the original equation: $(V_{\text{IL}} - V_{\text{T}})^2 / 2 = \left[ (V_{\text{DD}} - V_{\text{IL}} - V_{\text{T}}) \left( \frac{V_{\text{DD}}}{2} - V_{\text{IL}} \right) - \frac{\left( \frac{V_{\text{DD}}}{2} - V_{\text{IL}} \right)^2}{2} \right]$

Step 5: Expand and Simplify

Now expand the terms and simplify the equation to isolate $V_{\text{IL}}$. After expanding both sides, collect terms of $V_{\text{IL}}$ and solve.

Let me handle this algebraic expansion to get the resultant equation.The solution for $V_{\text{IL}}$ after substituting the second equation into the first is:

$V_{\text{IL}} = \frac{3V_{\text{DD}}}{8} + \frac{V_{\text{T}}}{4}$

Would you like a detailed breakdown of the steps involved in solving this, or do you have any other questions?

Here are 5 related questions:

1. How does the threshold voltage $V_T$ affect the value of $V_{\text{IL}}$?
2. What physical meaning does $V_{\text{IL}}$ represent in this context?
3. How would the equation change if the source voltage $V_{\text{DD}}$ were increased?
4. Can you derive the relationship between $V_{\text{OUT}}$ and $V_T$ using these equations?
5. How does this result apply to transistor behavior in digital circuits?

Tip: Expanding and simplifying equations early on can save a lot of time when solving for variables.