To complete the equation \(-7x - 12 = __ , x + 12\) so that it has no solution, we need to create a contradiction.
Let's start by rearranging the equation. The left side is \(-7x - 12\) and the right side is \( _ , x + 12\).
If we want there to be no solution, we want the left side and the right side to be equal for some values of \(x\), but then give a statement that is always false.
One way to do this is to set the terms such that the coefficients of \(x\) create a scenario where two linear equations are parallel.
For instance, if we make the coefficient of \(x\) in the blank (let's denote it as \(a\)) equal to \(-7\) to match the left side, then we also ensure the constant terms differ in a way that they can't equal each other.
Setting it up:
Let's use the equation \(a = -7\) but then make the constants such that: \[ -12 = 9 \]
This leads us to: -7x - 12 = -7x + 12.
Now, this becomes: \[ -7x - 12 = -7x + 12 \] Subtracting \(-7x\) from both sides, we get: \[ -12 = 12 \]
This is a contradiction, meaning no value of \(x\) can satisfy the equation.
Therefore, we fill in the blank with \(-7\) so that the equation reads:
\[ -7x - 12 = -7x + 12 \]
This equation has no solution.