Solve each of the following systems of homogeneous linear equations:

x + y – 2z = 0

2x + y – 3z = 0

5x + 4y – 9z = 0

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

x = k

similarly,

y = k

Hence, x = y = z = k.