Question

find the missing number so that the equation has no solutions. -2(-x + 10) = \square x + 15

Explanation:

Step1: Simplify left side

First, we simplify the left - hand side of the equation \(-2(-x + 10)\) using the distributive property \(a(b + c)=ab+ac\). Here \(a=-2\), \(b=-x\) and \(c = 10\). So \(-2(-x+10)=(-2)\times(-x)+(-2)\times10=2x - 20\). The equation becomes \(2x-20=\square x + 15\).

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 \(2x-20=\square x + 15\), we want the coefficient of \(x\) on the left - hand side to be equal to the coefficient of \(x\) on the right - hand side, and the constant terms \(- 20\) and \(15\) are already different (\(-20
eq15\)). So we set \(\square=2\). If we substitute \(\square = 2\) into the equation, we get \(2x-20 = 2x+15\). Subtracting \(2x\) from both sides gives \(-20=15\), which is a false statement, meaning the equation has no solutions.

Answer:

\(2\)