# The existence of the unique solution of the system for equations:

$

x + y + z = \lambda \\

5x - y + \mu z = 10 \\

2x + 3y - z = 6 \\

$

depends on

A. $\mu $ only

B. $\lambda $ only

C. $\lambda $ and $\mu $ both

D. neither $\lambda $ nor $\mu $

Last updated date: 26th Mar 2023

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Answer

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311.1k+ views

Hint: Express the given system of equations in matrix form and find the determinant of the coefficients of x,y and z.

We will write given equations in the matrix form as $A.X = B$

Where $A = \left( {\begin{array}{*{20}{c}}

1&1&1 \\

5&{ - 1}&\mu \\

2&3&{ - 1}

\end{array}} \right)$ , \[X = \left( {\begin{array}{*{20}{c}}

x \\

y \\

z

\end{array}} \right)\] and \[B = \left( {\begin{array}{*{20}{c}}

\lambda \\

{10} \\

6

\end{array}} \right)\]

Now, we will find determinant of A i.e. $\left| A \right|$

\[

\left| A \right| = \left( {\begin{array}{*{20}{c}}

1&1&1 \\

5&{ - 1}&\mu \\

2&3&{ - 1}

\end{array}} \right) \\

\left| A \right| = 1\left( {1 - 3\mu } \right) - 1\left( { - 5 - 2\mu } \right) + 1\left( {15 + 2} \right) \\

\left| A \right| = 1 - 3\mu + 5 + 2\mu + 17 \\

\left| A \right| = 23 - \mu \\

\]

From the above equation, we can see that the uniqueness of the system depends only on $\mu $.

$\therefore $Correct option is A.

Note: In a practical case, a system of linear equations will have a unique solution if the lines

representing the equations intersect each other at only one unique point i.e. the lines are

neither parallel nor coincident.

We will write given equations in the matrix form as $A.X = B$

Where $A = \left( {\begin{array}{*{20}{c}}

1&1&1 \\

5&{ - 1}&\mu \\

2&3&{ - 1}

\end{array}} \right)$ , \[X = \left( {\begin{array}{*{20}{c}}

x \\

y \\

z

\end{array}} \right)\] and \[B = \left( {\begin{array}{*{20}{c}}

\lambda \\

{10} \\

6

\end{array}} \right)\]

Now, we will find determinant of A i.e. $\left| A \right|$

\[

\left| A \right| = \left( {\begin{array}{*{20}{c}}

1&1&1 \\

5&{ - 1}&\mu \\

2&3&{ - 1}

\end{array}} \right) \\

\left| A \right| = 1\left( {1 - 3\mu } \right) - 1\left( { - 5 - 2\mu } \right) + 1\left( {15 + 2} \right) \\

\left| A \right| = 1 - 3\mu + 5 + 2\mu + 17 \\

\left| A \right| = 23 - \mu \\

\]

From the above equation, we can see that the uniqueness of the system depends only on $\mu $.

$\therefore $Correct option is A.

Note: In a practical case, a system of linear equations will have a unique solution if the lines

representing the equations intersect each other at only one unique point i.e. the lines are

neither parallel nor coincident.

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