Which classification best describes the following system of equations?12x+5y-3z=36x-2y+4z=39x-10y+5z=27inconsistent and dependentconsistent and dependentconsistent and independentinconsistent and independent

Answer: These three planes are consistent and independent. Explanation:Since,  if the system of planes has a solution then it is  called Consistent, While, if it does not have any solution then it is called inconsistent.Further, If the consistent system has infinite solution then it is dependent but if it has only a unique solution then it is called independent.Here, given equations of planes are 12x+5y-3z=36           -------(1 ) x-2y+4z=3                 -------(2) 9x-10y+5z=27          -------(3)From equation (1), 3z=12x+5y-36⇒z=4x+5y/3-12   ------(4)after putting this value in equation (2) and (3), we will get two equation in variables x and ySo, x-2y+4(4x+5y/3-12)=3 ⇒x-2y+16x+20y/3-48=3⇒3x-6y+48x+20y-144=9⇒51x+14y=153   --------(5)And, 9x-10y+5(4x+5y/3-12)=27⇒ 9x-10y+20x+25y/3-60=27⇒ 27x-30y+60x+25y-180=81⇒87x-5y=261   --------(6)after solving equation equation (5) and (6) we will get x=3 and y=0 substituting these values in equation (4) we will get z=0Thus the solution of these three plane (1), (2) and (3) is x=3, y=0 and z=oWhich is the unique solution, Thus the given planes are consistent and independent.