To solve this, we need to find values of $p$ and $q$ such that the given equation has infinitely many solutions.

The equation is:

$-7x + p = qx + 1$

For the equation to have infinitely many solutions, both the coefficients of $x$ and the constant terms must be equal on both sides. This is because two lines must coincide to have infinitely many solutions.

So, let's match terms:

1. The coefficient of $x$ on the left side is $-7$, and on the right side, it's $q$. Therefore: $-7 = q$

2. The constant term on the left side is $p$, and on the right side, it's $1$. Therefore: $p = 1$

Thus, the values are: $p = 1, \quad q = -7$

Would you like any further explanation?

Here are 5 related questions:

1. How can you identify when a system of linear equations has no solution?
2. What is the graphical interpretation of infinitely many solutions for a system of equations?
3. How do you determine if a system of equations is inconsistent?
4. Can a system of equations with different slopes ever have infinitely many solutions?
5. How do you solve a system of linear equations by substitution?

Tip: For infinitely many solutions, both the variable and constant terms must match exactly on both sides of the equation.