Question

Solve each of the following systems of homogeneous linear equations : 

3x + y + z = 0 

x – 4y + 3z = 0 

2x + 5y – 2z = 0

Answer 1

Given Equations :

3x + y + z = 0 

x – 4y + 3z = 0 

2x + 5y – 2z = 0

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

Now finding the Determinant of these set of equations,

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

= 3(8 – 15) – 1(– 2 – 6) + 1(5 + 8) 

= 3×(– 7) –1 × (– 8) + 1×13 

= – 21 + 8 + 13 = 0 

Since,

D = 0 

so the system of equation has infinite solution. 

Now, 

let z = k 

⇒ 3x + y = – k 

And,

x – 4y = – 3k 

Now using the cramer’s rule