If the system of linear equations 2x + 2y + 3 | vedclass

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

Question

(A) Given the system of equations:$2x + 2y + 3z = a$ $(1)$$3x - y + 5z = b$ $(2)$$x - 3y + 2z = c$ $(3)$For the system to have more than one solution,

Vedclass · Accepted Answer

$b - c - a = 0$

Vedclass · Answer

(A) Given the system of equations:$2x + 2y + 3z = a$ $(1)$$3x - y + 5z = b$ $(2)$$x - 3y + 2z = c$ $(3)$For the system to have more than one solution,the determinant of the coefficient matrix must be zero,and the system must be consistent.Let $D$ be the determinant of the coefficient matrix:$D = \begin{vmatrix} 2 & 2 & 3 \ 3 & -1 & 5 \ 1 & -3 & 2 \end{vmatrix} = 2(-2 + 15) - 2(6 - 5) + 3(-9 + 1) = 2(13) - 2(1) + 3(-8) = 26 - 2 - 24 = 0$.Since $D = 0$,the system has either no solution or infinitely many solutions.For infinitely many solutions,the equations must be linearly dependent. We observe that $(1) - (3) = (2x - x) + (2y - (-3y)) + (3z - 2z) = x + 5y + z = a - c$.Alternatively,notice that $(1) + (3) = 3x - y + 5z = a + c$. Comparing this with equation $(2)$,we get $a + c = b$,which implies $b - c - a = 0$.

If the system of linear equations $2x + 2y + 3z = a$,$3x - y + 5z = b$,and $x - 3y + 2z = c$,where $a, b, c$ are non-zero real numbers,has more than one solution,then:

Similar Questions

The system of equations $x+y+z=6$,$x+2y+5z=9$,$x+5y+\lambda z=\mu$ has no solution if

If the system of equations $kx + (k+1)y + (k-1)z = 0$,$(k-1)x + (k+2)y + kz = 0$,and $(k+1)x + ky + (k+2)z = 0$ has a non-trivial solution,then the sum of all possible values of $k$ is:

The number of values of $k$ for which the system of linear equations,$(k + 2)x + 10y = k$ and $kx + (k + 3)y = k - 1$ has no solution,is

If the solution for the system of equations $x+2y-z=3$,$3x-y+2z=1$ and $2x-2y+3z=2$ is $(\alpha, \beta, \gamma)$,then $\alpha^2+\beta^2+\gamma^2=$

The number of $3 \times 3$ matrices $A$,whose entries are either $1$ or $-1$ and for which the system $A\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix}$ has exactly three distinct solutions,is

Vedclass Test Series

Exam Paper Generator

Online Exam Module