Choice D is correct. Let $l$ represent the cost, in dollars, of $1$ loaf of bread, and let $d$ represent the cost, in dollars, of $1$ dozen eggs. It’s given that the first customer paid $12.45$ dollars for $1$ loaf of bread and $2$ dozen eggs. Therefore, the first customer’s purchase can be represented by the equation $l + 2 d = 12.45$ . It’s also given that the second customer paid $19.42$ dollars for $4$ loaves of bread and $1$ dozen eggs. Therefore, the second customer’s purchase can be represented by the equation $4 l + d = 19.42$ . The equations $l + 2 d = 12.45$ and $4 l + d = 19.42$ form a system of linear equations, which can be solved by elimination to find the value of $d$ . Multiplying the first equation in the system by $negative 4$ yields $- 4 l - 8 d = - 49.8$ . Adding $- 4 l - 8 d = - 49.8$ to the second equation, $4 l + d = 19.42$ , yields $(- 4 l + 4 l) + (- 8 d + d) = (- 49.8 + 19.42)$ , which is equivalent to $- 7 d = - 30.38$ . Dividing both sides of this equation by $negative 7$ yields $d equals 4.34$ . Therefore, the cost, in dollars, of $1$ dozen eggs is $4.34$ .

Choice A is incorrect. This is the cost, in dollars, of $1$ loaf of bread.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.