If the system of linear equations x + y + z = | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) For a system of linear equations to have infinitely many solutions,the third equation must be a linear combination of the first two equations. Let

Vedclass · Accepted Answer

$10$

Vedclass · Answer

(C) For a system of linear equations to have infinitely many solutions,the third equation must be a linear combination of the first two equations. Let the third equation be $L_3 = a L_1 + b L_2$. $x + 3y + \lambda z = \mu = a(x + y + z - 5) + b(x + 2y + 2z - 6)$. Comparing the coefficients of $x, y, z$ and the constant term: $a + b = 1$ (coefficient of $x$) $a + 2b = 3$ (coefficient of $y$) $a + 2b = \lambda$ (coefficient of $z$) $5a + 6b = \mu$ (constant term) Solving the first two equations: Subtracting $(a + b = 1)$ from $(a + 2b = 3)$ gives $b = 2$. Substituting $b = 2$ into $a + b = 1$ gives $a = -1$. Now,find $\lambda$ and $\mu$: $\lambda = a + 2b = -1 + 2(2) = 3$. $\mu = 5a + 6b = 5(-1) + 6(2) = -5 + 12 = 7$. Therefore,$\lambda + \mu = 3 + 7 = 10$.

If the system of linear equations $x + y + z = 5$,$x + 2y + 2z = 6$,and $x + 3y + \lambda z = \mu$ (where $\lambda, \mu \in \mathbb{R}$) has infinitely many solutions,then the value of $\lambda + \mu$ is:

Similar Questions

Identify the correct statement:

If $A$ is a matrix such that $\left[\begin{array}{ll} 2 & 1 \\ 3 & 2 \end{array}\right] A \left[\begin{array}{l} 1 \\ 1 \end{array}\right] = \left[\begin{array}{l} 1 \\ 0 \end{array}\right]$,then $A$ is equal to

If the system of linear equations: $x + y + z = 6, x + 2y + 5z = 10, 2x + 3y + \lambda z = \mu$ has infinitely many solutions,then the value of $\lambda + \mu$ equals:

If $\begin{bmatrix} 2 & 1 & 1 \\ 0 & 3 & -1 \\ 1 & -1 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix}$,then $\begin{bmatrix} x \\ y \\ z \end{bmatrix} =$

For $\alpha, \beta \in R$,suppose the system of linear equations $x-y+z=5$,$2x+2y+\alpha z=8$,and $3x-y+4z=\beta$ has infinitely many solutions. Then $\alpha$ and $\beta$ are the roots of:

Vedclass Test Series

Exam Paper Generator

Online Exam Module