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Solve the system of homogeneous linear equations

x + y – 2z = 0

2x + y – 3z =0

5x + 4y – 9z = 0

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Given as x + y – 2z = 0

2x + y – 3z =0

5x + 4y – 9z = 0

The system of any equation can be written in matrix form as AX = B

Finding the determinant of these set of equations,

= 1(1 × (– 9) – 4 × (– 3)) – 1(2 × (– 9) – 5 × (– 3)) – 2(4 × 2 – 5 × 1)

= 1(– 9 + 12) – 1(– 18 + 15) – 2(8 – 5)

= 1 × 3 –1 × (– 3) – 2 × 3

= 3 + 3 – 6

= 0

Here D = 0, therefore the system of equation has infinite solution.

Let z = k

⇒ x + y = 2k

And 2x + y = 3k

On using the Cramer’s rule

x = k

For D2/D

y = k

Hence x = y = z

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