Final answer:
The value that must go in the blank for the equation to have infinitely many solutions is -21, since it would make both sides of the equation identical, confirming that there are infinitely many solutions.
Step-by-step explanation:
To determine the value that must go in the blank for the equation to have infinitely many solutions, we need to make the equation true for all values of x. The original equation is 7(x - 3) - 2x = 5x + __.
First, let's simplify the left side of the equation by distributing the 7 and combining like terms:
- 7(x - 3) - 2x = 7x - 21 - 2x
- 7x - 2x - 21 = 5x - 21
- 5x - 21 = 5x - 21
For this equation to have infinitely many solutions, the expressions on both sides of the equation must be identical. As we can see after simplifying, the identical term on both sides should be -21. Therefore, the blank should be filled with -21 so that:
7(x - 3) - 2x = 5x - 21
This gives us the equation 5x - 21 = 5x - 21, which is true for all values of x, confirming that there are infinitely many solutions.