Question

How do you find the missing number so that the equation has no solutions $5x - 11 = ?x + 19$?

Answer 1

For the equation to not have solution, it should fail the distributive law which is
$LHS \ne RHS$or in other words eliminate the “x”. To fail the distributive law, we have to prove that $LHS \ne RHS$,to do this we need to get rid of the variable that exists in the equations. In that process, we will be able to find the missing number as well.

Complete step by step solution:
First, we start to bring all the variables on one side and the constants to one side.
This can be done by subtracting ?x on both sides first, which implies
$5x - 11 - ?x = ?x + 19 - ?x$
Then we have
$\Rightarrow 5x - 11 - ?x = 19$
Further, we have to add 11 on both sides
$\Rightarrow 5x - 11 - ?x + 11 = 19 + 11$
Further simplification we get
$\Rightarrow 5x - ?x = 30$
We can observe here that x is common on the left-hand side of the equation. We take common and get;
$\left( {5 - ?} \right)x = 30$
As we have been asked to show the equation has no solution, the missing number has to be in such a way that the variable x gets eliminated leaving no variable to get a solution. So, consider;
$\Rightarrow 5 - ? = 0$ then
$\Rightarrow ? = 5$
Then, we substitute the value of ? and we eliminate x
$\Rightarrow (5 - 5)x = 30 \\\ \Rightarrow 0\times x = 30 \\\ \Rightarrow 0 = 30 \\\$
Which is obviously not true.
This false statement, clearly proves that there does not exist a solution for the given equation

Hence, we have found the missing number with the equation having no solution.

Note: For a solution to exist, the variable should not get eliminated by any means and since the question demanded for the equation to have no solution, we eliminated the variable “x”. Also, the equation shall not only have no solutions but also maybe it can have infinitely many solutions which fit in a certain pattern.