Final answer:
To give the equation 25x + __ = 73 - 25x infinitely many solutions, the blank should be filled with 73, resulting in an identical expression on both sides of the equation.
Step-by-step explanation:
To have an equation with infinitely many solutions, both sides of the equation must be identical after all simplifications. For the given equation 25x + __ = 73 - 25x, we add 25x to both sides to achieve this balance:
25x + 25x = 73 - 25x + 25x
50x = 73
Therefore, the equation with infinitely many solutions would actually require a different approach. We want both sides of the equation to be the same when simplified. We can do this by making sure the term added to 25x is equivalent to 73. So, the blank should be filled with 73 to yield:
25x + 73 = 73 - 25x.
Now, when we add 25x to both sides, we get:
25x + 25x + 73 = 73 + 25x + 25x
50x + 73 = 73 + 50x.
Simplifying further:
50x + 73 = 50x + 73.
As both sides are identical, the original equation 25x + 73 = 73 - 25x has infinitely many solutions.