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Solve each of the following system of homogeneous linear equations:2x+3y+4z=0 x+y+z=0 2x+5y-2z=0
CBSE | Class 12 | Determinants | Exercise 6.5 | Maths | RD Sharma
Solve each of the following system of homogeneous linear equations:2x+3y+4z=0 x+y+z=0 2x+5y-2z=0

According to the question,

2x+3y+4z=0

x+y+z=0

2x+5y-2z=0

Any equation can be written in matrix form as AX=B

Now find the Determinant of these set of equations,

D=\left| \begin{matrix} 2 & 3 & 4 \\ 1 & 1 & 1 \\ 2 & 5 & -2 \\ \end{matrix} \right|

\left| A \right|=2\left| \begin{matrix} 1 & 1 \\ 5 & -2 \\ \end{matrix} \right|-3\left| \begin{matrix} 1 & 1 \\ 2 & -2 \\ \end{matrix} \right|+4\left| \begin{matrix} 1 & 1 \\ 2 & 5 \\ \end{matrix} \right|

=2\left( 1\times \left( -2 \right)-1\times 5 \right)-3\left( 1\times \left( -2 \right)-2\times 1 \right)+4\left( 1\times 5-2\times 1 \right)

=2\left( -2-5 \right)-3\left( -2-2 \right)+4\left( 5-2 \right)

=1\times \left( -7 \right)-3\times \left( -4 \right)+4\times 3

=-7+12+12
=17

Thus, D\ne 0, so the system of equation has infinite solution.

Hence, the system of equation has only solution as x=y=z=0.

i

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