System of equations: { 5x - y = 5 { y - 5x + 5 = 0

Analysis of the system of equations:

The given system consists of two linear equations. Let's analyze them to determine the nature of their solutions.

Equation 1:

The first equation is \( 5x - y = 5 \).

Equation 2:

The second equation is \( y - 5x + 5 = 0 \).

Step-by-step solution:

1. Rewrite Equation 2 in a standard form:
   To make comparison easier, let's rearrange the second equation to the form \( Ax + By = C \).
   From \( y - 5x + 5 = 0 \), we can add \( 5x \) and subtract \( 5 \) from both sides:
   \( y = 5x - 5 \)
   Now, rearrange it to match the first equation's structure:
   \( -5x + y = -5 \)
2. Compare the two equations:
   Equation 1: \( 5x - y = 5 \)
   Equation 2 (rearranged): \( -5x + y = -5 \)
   Let's multiply Equation 2 by -1 to make the coefficients match the first equation more closely:
   \( -1 · (-5x + y) = -1 · (-5) \)
   This gives us \( 5x - y = 5 \).
3. Analyze the result:
   We observe that both equations simplify to the same equation: \( 5x - y = 5 \). This means the two equations are dependent and represent the same line.

Conclusion:

Since both equations describe the same line, any point on this line is a solution to the system. Therefore, the system has infinitely many solutions.

Answer: The system has infinitely many solutions.