The missing number is -20.
To have no solutions, the equation must be equivalent to a statement of the form 0 = a, where a is a non-zero constant.
Let's solve the equation for x:
1. Subtract 3 from both sides:
x + 3 - 3 = 5x + 19 - 3
x = 5x + 16
2. Subtract 5x from both sides:
x - 5x = 5x + 16 - 5x
-4x = 16
3. Divide both sides by -4:
-4x/-4 = 16/-4
x = -4
Now, substitute -4 for x in the original equation:
-4 + 3 = 5(-4) + 19
-1 = -20 + 19
-1 = -1
This is a true statement. However, for the equation to have no solutions, the resulting statement needs to be 0 = a, where a is a non-zero constant.
To achieve this, we need to add an additional term to the equation that will cancel out the existing terms. This term needs to have a coefficient equal to the negative of the coefficient of x in the original equation (5) and an opposite sign on the constant term.
Therefore, we need to add a term of -5x - 20. This will cancel out the 5x in the equation and leave us with:
-1 - 5x - 20 = 0
This equation satisfies the condition for having no solutions, as it is now equivalent to the statement 0 = -21, which is a true statement with a non-zero constant.