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  • Examine the consistency of the system of equations x + y + z = 1, 2x + 3y + 2z =2, ax + ay + 2az = 4

Q : 4         Examine the consistency of the system of equations.

                 \small x+y+z=1

                 \small 2x+3y+2z=2

                 \small ax+ay+2az=4

Answers (1)

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We have given the system of equations:

                                   \small x+y+z=1

                                 \small 2x+3y+2z=2

                                 \small ax+ay+2az=4

The given system of equations can be written in the form of the matrix; AX =B

where A = \begin{bmatrix} 1& 1&1 \\ 2& 3& 2\\ a& a &2a \end{bmatrix},  X = \begin{bmatrix} x\\y \\ z \end{bmatrix}  and B = \begin{bmatrix} 1\\2 \\ 4 \end{bmatrix}.

So, we want to check for the consistency of the equations;

|A| = 1(6a-2a) -1(4a-2a)+1(2a-3a)

= 4a -2a-a = 4a -3a =a \neq 0 

[If zero then it won't satisfy the third equation]

Here A is non- singular matrix therefore there exist A^{-1}.

Hence, the given system of equations is consistent.

Posted by

Divya Prakash Singh

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