For the system of linear equations $2x + 3y + 5z = 9$, $7x + 3y - 2z = 8$and $2x + 3y + \lambda z = \mu $. Under what condition does the above system of equations have infinitely many solutions.
A) $\lambda = 5$ and $\mu \ne 9$
B) $\lambda = 5$ and $\mu = 9$
C) $\lambda = 9$ and $\mu \ne 5$
D) $\lambda = 9$ and $\mu = 5$

Question

For the system of linear equations $2x + 3y + 5z = 9$, $7x + 3y - 2z = 8$and $2x + 3y + \lambda z = \mu $. Under what condition does the above system of equations have infinitely many solutions.
A) $\lambda = 5$ and $\mu \ne 9$
B) $\lambda = 5$ and $\mu = 9$
C) $\lambda = 9$ and $\mu \ne 5$
D) $\lambda = 9$ and $\mu = 5$

Vedantu Content Team · Accepted Answer

Hint:When the numbers of variables are greater than the number of equations then the system of linear equations has infinitely many solutions. Try to make two equations the same and evaluate the value of $\lambda $and $\mu $.Complete step-by-step answer:We are given a system of linear equations $2x + 3y + 5z = 9$, $7x + 3y - 2z = 8$and $2x + 3y + \lambda z = \mu $.First, we give numbers to each equation.Let,$2x + 3y + 5z = 9....(1)$$7x + 3y - 2z = 8....(2)$$2x + 3y + \lambda z = \mu ....(3)$We have to find the value of $\lambda $and for which the above system of linear equations have infinitely many solutions$\mu $ions.We know that when the numbers of variables are greater than the number of equations then the system of linear equations has infinitely many solutions.But there are three variables and three equations, so we make two equations the same so that numbers of variables are greater than the number of equations.If we see equation $(1)$ and $(3)$ the coefficients of $x$ and $y$ are the same, we have to make the same coefficients of $z$and constant.Substitute $\lambda  = 5$ and $\mu  = 9$ in equation $(3)$.$2x + 3y + 5z = 9....(3)$Now equations $(1)$ and $(3)$ are the same. It means we have two equations and three variables.Therefore, $\lambda  = 5$ and $\mu  = 9$ is the correct answer.Hence, option (B) is correct.Note:When the system of linear equations has infinite solutions, it means two or more of the planes are parallel or we can say that if the three planes end up lying on the top of each other. Parallel planes have the same equations with the same variable, they just differ by the constant values. If the system has infinite solutions then we also called it consistent.