Find the missing number so that the equation has no solutions. \[ \square x+19=-2 x+14 \]

To determine the missing number \( a \) in the equation: \[ a x + 19 = -2x + 14 \] such that the equation has **no solutions**, follow these steps: 1. **Rearrange the Equation:** Move all terms involving \( x \) to one side: \[ a x + 2x = 14 - 19 \] \[ (a + 2)x = -5 \] 2. **Analyze for No Solutions:** For the equation to have no solutions, the coefficient of \( x \) must be zero while the constant term is non-zero. This results in a contradiction: \[ 0 \cdot x = -5 \] This is impossible since \( 0 \neq -5 \). 3. **Find \( a \) When \( a + 2 = 0 \):** \[ a + 2 = 0 \] \[ a = -2 \] When \( a = -2 \), the original equation becomes: \[ -2x + 19 = -2x + 14 \] Subtracting \(-2x\) from both sides: \[ 19 = 14 \] This is a contradiction, meaning there's no solution. **Answer:** \(-2\)