To solve this system of linear equations and determine how many solutions it has, we begin by analyzing the two given equations:

1. $8x - 4y = 24$
2. $16x - 8y = 48$

Step 1: Check if the second equation is a multiple of the first

We can see that the second equation looks like a multiple of the first equation. Let's divide the second equation by 2:

$\frac{16x - 8y = 48}{2} \implies 8x - 4y = 24$

This results in the same equation as the first one. Therefore, the second equation is not independent from the first but rather a scalar multiple of it.

Step 2: Conclusion about the solutions

Since both equations represent the same line, every solution to the first equation is also a solution to the second equation. This means the system has infinitely many solutions because the two equations describe the same line.

Final Answer:

The system of equations has infinitely many solutions.

Would you like further clarification on any part of the process or details on how to interpret this result? Here are five related questions you could explore:

1. How do we identify if two lines are parallel, identical, or intersecting from their equations?
2. What is the general method to solve a system of linear equations with substitution or elimination?
3. How can graphing help in understanding the solution of systems of equations?
4. What are the different types of solutions (no solution, unique solution, infinite solutions) for linear systems?
5. How can we represent systems of equations in matrix form and solve them using Gaussian elimination?

Tip: When solving systems of equations, checking if one equation is a multiple of another is a quick way to identify dependent or inconsistent systems.