The problem presents a pair of simultaneous equations:

$2x + 2y = 1$ $3^{2x + 4y} = 27$

Let's solve this step by step.

Step 1: Simplify the Second Equation

The second equation can be written as: $3^{2x + 4y} = 3^3$ Since the bases are the same, we can equate the exponents: $2x + 4y = 3$

So now we have two equations:

1. $2x + 2y = 1$
2. $2x + 4y = 3$

Step 2: Solve the System of Equations

Subtract the first equation from the second to eliminate $2x$: $(2x + 4y) - (2x + 2y) = 3 - 1$ $2y = 2$ $y = 1$

Step 3: Substitute $y = 1$ into the First Equation

Now, substitute $y = 1$ into the first equation: $2x + 2(1) = 1$ $2x + 2 = 1$ $2x = -1$ $x = -\frac{1}{2}$

Solution

The solution to the system is: $x = -\frac{1}{2}, \quad y = 1$

Would you like a detailed explanation of any step?

Here are five related questions for further exploration:

1. How would the solution change if the constant in the second equation was different?
2. Can you solve a similar system if the second equation used a different base, such as $2$ instead of $3$?
3. What is the graphical interpretation of these solutions?
4. How do you verify that the solution satisfies both original equations?
5. Can this problem be solved using matrix methods?

Tip: When solving systems of equations, try to eliminate one variable to simplify the process.