Final answer:
The missing number in Luisa's equation must be 56, as this creates identical expressions on both sides, resulting in an equation with infinite solutions where 7x + 21 equals 7x + 21.
Step-by-step explanation:
Luisa is working on an algebraic equation, 7(x + 3) = 7(x - 5), and the page is torn, missing one number. She knows the equation has infinite solutions, which means both sides of the equation must be identical for any value of x. Since the coefficients of x are the same on both sides, the constants must also be the same once the number is known.
To find the missing number, we simplify the equation by applying the distributive property: 7x + 21 = 7x - 35. For the equation to have infinite solutions, the constant terms on both sides must be equal, which currently they are not. So, we must assume the missing number changes the -35 so that it equals +21.
The missing number, therefore, must be a +6 since -35 + 6 = -29, which is not equal to 21. To have infinite solutions, the number has to be 56, making the equation 7x + 21 = 7x + 21, satisfying the condition for infinite solutions. Thus, we can eliminate terms and find that the missing number is 56.