For the continuum and discrete elastic equations, we derive exact artificial boundary conditions (ABCs), often referred to as transparent boundary conditions, that can be applied at a planar interface below which there are no forces. Solution of the elasticity equations can then be performed using this interface as an artificial boundary, often with greatly reduced computational effort, but without loss of accuracy. A general solvability requirement is presented for the existence of an artificial boundary operator for discrete systems (such as discrete elasticity) on an unbounded (semi-infinite) domain. The solvability requirement is validated by introducing a sum-of-exponentials ansatz for the solution below the artificial boundary. We also derive a new expression for the total energy for the system, involving only the region above the artificial boundary. Numerical examples are provided to confirm and illustrate the accuracy and effectiveness of the results.