From the provided image, the question is about solving the system of equations by substitution and determining whether the system is consistent, inconsistent, or has dependent equations. Here's a detailed explanation:

Given System:

1. $y = 4x$
2. $16x - 4y = 0$

Step 1: Substitution

Since $y = 4x$, substitute $y$ into the second equation: $16x - 4(4x) = 0$

Simplify: $16x - 16x = 0$

This simplifies to $0 = 0$, which is always true.

Step 2: Analyze the Result

The equation $0 = 0$ indicates that the two equations are dependent; they represent the same line. Hence, there are infinitely many solutions.

Solution Set:

The solution set is all points that satisfy $y = 4x$. In set notation: $\{(x, y) \mid y = 4x\}$

Correct Answer:

Option B: There are infinitely many solutions, and the equations are dependent. The solution set is $\{(x, y) \mid y = 4x\}$.

Would you like additional details on solving systems of equations or exploring similar problems?

Relative Questions:

1. How can you identify dependent equations in a system?
2. What methods besides substitution can solve systems of linear equations?
3. What is the difference between consistent and inconsistent systems?
4. How do you graphically represent a system with infinitely many solutions?
5. What are the practical applications of solving systems with dependent equations?

Tip:

When solving systems of equations, always simplify and substitute carefully. Dependent systems often simplify to a tautology like $0 = 0$.