Final answer:
The missing number to ensure the equation has infinitely many solutions is 19. This is found by making the coefficients of x and the constant terms on both sides of the equation equal when simplified.
Step-by-step explanation:
To find the missing number that allows the equation to have infinitely many solutions, we need to ensure both sides of the equation are identical when simplified. Let's first simplify both sides of the given equation:
- 3(-5x + 17) = 2(2x + 16) ?x + 19
- -15x + 51 = 4x + 32 ?x + 19
For the equation to have infinitely many solutions, the coefficients of x and the constants on both sides must be equal. This means we must find the value of '?' that satisfies the following equation:
- -15x + ?x = 4x (For the x coefficients to be equal)
- 51 = 32 + 19 (For the constants to be equal)
Now, solving for '?':
- -15 + ? = 4
- ? = 4 + 15
- ? = 19
Therefore, the missing number is 19 to ensure that the equation has infinitely many solutions.