# Solve each of the following systems of homogeneous linear equations : x + y – 2z = 0  2x + y – 3z = 0  5x + 4y – 9z = 0

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Solve each of the following systems of homogeneous linear equations :

x + y – 2z = 0

2x + y – 3z = 0

5x + 4y – 9z = 0

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Given Equations :

x + y – 2z = 0

2x + y – 3z = 0

5x + 4y – 9z = 0

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

Now 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

Since,

D = 0

so the system of equation has infinite solution.

Now,

let z = k

⇒ x + y = 2k

And,

2x + y = 3k

Now,

using the cramer’s rule

Hence,

x = y = z = k.