RTUComputer ScienceYr 2024 · Sem 32024

Q8Advanced Engineering Mathematics

Question

2 marks

Find all the basic solution of the system: 2 x + y - z = 2 3 x + 2 y + z = 3

Answer

Set each variable to zero to find basic solutions for the underdetermined linear system.

System of equations: $2 x + y - z = 2$ and $3 x + 2 y + z = 3$ .

We have 2 linear equations and 3 variables. The maximum number of basic solutions is found by setting $n - m = 3 - 2 = 1$ variable to zero at a time.

Set $x = 0$ : $y - z = 2$ , $2 y + z = 3 ⟹ 3 y = 5 ⟹ y = 5/3, z = - 1/3$ . Basic Solution: $(0, 5/3, - 1/3)$
Set $y = 0$ : $2 x - z = 2$ , $3 x + z = 3 ⟹ 5 x = 5 ⟹ x = 1, z = 0$ . Basic Solution: $(1, 0, 0)$
Set $z = 0$ : $2 x + y = 2$ , $3 x + 2 y = 3$ . Multiplying first by 2: $4 x + 2 y = 4$ . Subtracting gives $x = 1, y = 0$ . Basic Solution: $(1, 0, 0)$