Abstract: An $A_{\infty}$-algebra generalizes an associative algebra, by requiring the binary operation to only be associative up to the derivative of a ternary operation. There is then a 4-ary operation satisfying some relation between these two, and we can continue to get the structure of an operad. Beginning with an augmented differential graded associative (dga) algebra, we use the Eilenberg-MacLane bar construction to get a dga coalgebra with some nice properties. We use this to define a general $A_{\infty}$-algebra, and then manipulate it as a chain complex to find the homotopy.