Let's first break down the problem so we can develop a solution for it.

We have the following:

1. Writing the core gameplay code requires 7 hours per feature.
2. Implementing AI algorithms takes 5 hours per feature.
3. Integrating audio takes 2 hours per feature.
4. A total of 46 hours are available for these tasks.
5. The group needs to complete 12 features in total.
6. The number of AI features (y) is twice the number of core gameplay features (x).
7. Let:
   • $x$ = number of core gameplay features,
   • $y$ = number of AI features,
   • $z$ = number of audio features.

From the information given, we can derive the following system of equations:

$1. 7x + 5y + 2z = 46 \quad \text{(time constraint)}$ $2. x + y + z = 12 \quad \text{(total number of features)}$ $3. y = 2x \quad \text{(AI features are twice the core gameplay features)}$

We'll now solve this system of equations using RStudio. Here's the R code to solve it:

rCopy code
# Define the coefficients of the system of equations
library(matlib)

# Coefficients matrix
A <- matrix(c(7, 5, 2, 1, 1, 1, 1, -2, 0), nrow = 3, byrow = TRUE)

# Constants matrix
b <- c(46, 12, 0)

# Solve the system of equations
solution <- solve(A, b)
solution

# The result is the values of x, y, and z
x <- solution[1]  # Core gameplay features
y <- solution[2]  # AI features
z <- solution[3]  # Audio integration features

# Output the solution
cat("Number of core gameplay features (x):", x, "\n")
cat("Number of AI features (y):", y, "\n")
cat("Number of audio integration features (z):", z, "\n")

This code solves the system of linear equations to find $x$, $y$, and $z$, which represent the number of core gameplay features, AI features, and audio integration features, respectively.

Do you want further explanation or details on how this works? Here are five related questions that build upon this:

1. How do matrix operations like inversion help in solving linear equations?
2. What would happen if the relationship between the features changed (e.g., $y = 3x$)?
3. Can you generalize this problem for different project time constraints?
4. How can optimization techniques like linear programming be used here to optimize feature development?
5. How can we use this R code to analyze other similar problems with different constraints?

Tip: Always ensure that your system of equations is consistent and independent before attempting to solve it using matrix methods like inversion.