Question

Find the missing number so that the equation has no solutions. _x–3= – 4( – 2x–1)–3x–2

Answer 1

To determine the missing number in the equation \( _x - 3 = -4(-2x - 1) - 3x - 2 \) so that the equation has no solutions, we first need to simplify the equation.

1. **Start by simplifying the right side**:

\[
-4(-2x - 1) = 8x + 4
\]

Now substitute this back into the equation:

\[
_x - 3 = 8x + 4 - 3x - 2
\]

Combine the terms on the right:

\[
8x - 3x + 4 - 2 = 5x + 2
\]

So, the equation simplifies to:

\[
_x - 3 = 5x + 2
\]

2. **Rearranging the equation**:

Move \( 5x \) to the left side:

\[
_x - 5x - 3 = 2
\]

Combine the \( x \)-terms:

\[
(_ - 5)x - 3 = 2
\]

3. **Solving for \( x \)**:

To isolate \( x \), add 3 to both sides:

\[
(_ - 5)x = 2 + 3
\]
\[
(_ - 5)x = 5
\]

Now, to find \( x \), divide both sides by \( (_ - 5) \):

\[
x = \frac{5}{_ - 5}
\]

4. **Condition for no solutions**:

The equation has no solutions when the coefficient of \( x \) is zero (that is, there is no variable term). This happens when:

\[
_ - 5 = 0
\]

Solving for \( _ \):

\[
_ = 5
\]

Therefore, the missing number is \( \boxed{5} \).