Final Answer:
The values of (m) and (n) that will make the system consistent and independent are (m=5) and (n=3). Substituting these values into the system of equations yields a unique solution, satisfying both conditions for consistency and independence.Thus,the correct option is c.
Step-by-step explanation:
In the given system, with equations (2x + my = 8) and (4x + ny = 15), we seek values for (m) and (n) that yield a consistent and independent solution. Opting for (m=5) and (n=3) transforms the system into (2x + 5y = 8) and (4x + 3y = 15). Through algebraic manipulation or matrix methods, solving this system demonstrates the existence of a unique solution for (x) and (y), signifying consistency.
Moreover, the key to establishing independence lies in the coefficients of (y). The coefficients 5 and 3 in the respective equations are distinct, indicating that the equations are not scalar multiples of each other. This lack of proportionality underscores the independence of the system's equations. Therefore, by choosing (m=5) and (n=3), we ensure both the existence of a solution and the non-dependence of the equations, meeting the criteria for a consistent and independent system.
Therefore,the correct option is c.