Given equation:
4x + 5y = 2(2p + 7q)x + (p + 8q)y - 2q - p + 1 = 0

Infinitely many solutions:
For a linear equation to have infinitely many solutions, the two equations must represent the same line. This means that the coefficients of x and y in the two equations must be proportional.

Coefficients of x:
From the given equation, we have:
4x = 4(2p + 7q)x
Comparing the coefficients of x, we get:
4 = 4(2p + 7q)
1 = 2p + 7q

Coefficients of y:
From the given equation, we have:
5y = (p + 8q)y - 2q - p + 1
Comparing the coefficients of y, we get:
5 = p + 8q

Solving the equations:
We now have two equations:
1 = 2p + 7q
5 = p + 8q

We can solve these equations simultaneously to find the values of p and q.

Method 1: Substitution method
- Solve one equation for one variable in terms of the other variable.
- Substitute this expression into the other equation and solve for the remaining variable.
- Substitute the value of the remaining variable back into one of the original equations to find the other variable.

Method 2: Elimination method
- Multiply one or both equations by appropriate constants to make the coefficients of one variable in both equations equal.
- Add or subtract the equations to eliminate one variable and solve for the other variable.
- Substitute the value of the remaining variable back into one of the original equations to find the other variable.

Summary:
To find the values of p and q for which the linear equation has infinitely many solutions, we need to solve the equations:
1 = 2p + 7q
5 = p + 8q

Using either the substitution or elimination method, we can find the values of p and q that satisfy both equations. These values will result in the given equation representing the same line, indicating infinitely many solutions.