Question

find the missing number so that the equation has no solutions. 2(-2x + 5) = \square x - 12

Explanation:

Step1: Simplify left side

First, distribute the 2 on the left side of the equation. Using the distributive property \(a(b + c)=ab+ac\), we have:
\(2(-2x + 5)=2\times(-2x)+2\times5=-4x + 10\)
So the equation becomes \(-4x + 10=\square x-12\)

Step2: Analyze no - solution condition

For a linear equation of the form \(ax + b=cx + d\) to have no solutions, the coefficients of \(x\) must be equal (\(a = c\)) and the constant terms must be different (\(b
eq d\)).
In our equation \(-4x + 10=\square x-12\), we want the coefficient of \(x\) on the right - hand side to be equal to the coefficient of \(x\) on the left - hand side. The coefficient of \(x\) on the left is \(- 4\), so the missing number (the coefficient of \(x\) on the right) should be \(-4\). Let's verify:
If the missing number is \(-4\), the equation is \(-4x + 10=-4x-12\)
If we add \(4x\) to both sides, we get \(10=-12\), which is a false statement. So the equation has no solutions when the missing number is \(-4\).

Answer:

\(-4\)