Final answer:
Equations a (-19x - 38 = -19x - 38) and d (-19x + 91 = -19x + 91) have infinitely many solutions because both sides are identical, satisfying the equation for any value of x.
Step-by-step explanation:
The question deals with identifying which equations from the given options have infinitely many solutions. An equation has infinitely many solutions when both sides of the equation are identical, meaning every value of the variable satisfies the equation.
Options a (-19x - 38 = -19x - 38) and d (-19x + 91 = -19x + 91) are identical on both sides, therefore they have infinitely many solutions. This is because, regardless of the value of x, subtracting or adding the same quantity to it will always yield a true statement.
However, options b (-19x - 19 = -19x - 19) and c (-19x – 19 = -19x + 19) do not have infinitely many solutions because they are not identical on both sides of the equation when simplified.