To determine the values of $a$ and $b$ for which the given system of linear equations has infinitely many solutions, we need to make sure that the two equations are dependent, meaning they represent the same line.

The given system of equations is:

1. $2x - (a - 4)y = 2b + 1$
2. $4x - (a - 1)y = 5b - 1$

First, we rewrite both equations in the standard form $ax + by + c = 0$.

1. The first equation: $2x - (a - 4)y - (2b + 1) = 0$

   Here, the coefficients are: $a_1 = 2, \quad b_1 = -(a - 4), \quad c_1 = -(2b + 1)$

2. The second equation: $4x - (a - 1)y - (5b - 1) = 0$

   Here, the coefficients are: $a_2 = 4, \quad b_2 = -(a - 1), \quad c_2 = -(5b - 1)$

For the system to have infinitely many solutions, the ratios of the corresponding coefficients must be equal:

$\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$

Substituting the values:

$\frac{2}{4} = \frac{-(a - 4)}{-(a - 1)} = \frac{-(2b + 1)}{-(5b - 1)}$

Let's solve these equations:

1. Solving $\frac{2}{4} = \frac{-(a - 4)}{-(a - 1)}$:

$\frac{1}{2} = \frac{a - 4}{a - 1}$

Cross-multiplying gives:

$a - 4 = \frac{1}{2}(a - 1)$

Multiply through by 2 to eliminate the fraction:

$2(a - 4) = a - 1$

Expanding and solving for $a$:

$2a - 8 = a - 1$

$2a - a = 8 - 1$

$a = 7$

2. Solving $\frac{-(a - 4)}{-(a - 1)} = \frac{-(2b + 1)}{-(5b - 1)}$:

Since we already know $a = 7$, substitute this into the equation:

$\frac{-(7 - 4)}{-(7 - 1)} = \frac{-(2b + 1)}{-(5b - 1)}$

Simplifying:

$\frac{-3}{-6} = \frac{-(2b + 1)}{-(5b - 1)}$

$\frac{1}{2} = \frac{2b + 1}{5b - 1}$

Cross-multiplying gives:

$5b + 1 = 4b + 2$

Solving for $b$:

$5b - 4b = 2 - 1$

$b = 1$

Final Answer:

The system of linear equations will have infinitely many solutions for $\boxed{a = 7}$ and $\boxed{b = 1}$.

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Would you like more details or have any questions? Here are some related questions to consider:

1. How do the coefficients influence the number of solutions in a system of equations?
2. What happens if the value of $a$ or $b$ changes slightly?
3. Can we find a unique solution for different values of $a$ and $b$?
4. How can the concept of determinants help in analyzing this system?
5. What does it mean for two lines to be dependent geometrically?

Tip: Always check each ratio when determining conditions for infinite solutions; any inconsistency can indicate no solutions instead.