Question

Let S be the set of all column matrices $\left[\begin{array}{ccc}b1\\b2\\b3\end{array}\right]$ such that b₁, b₂, b₃ ∊ R and the system of equations (in real
variables)
-x + 2y + 5z = b₁
2x - 4y + 3z = b₂
x -2y + 2z = b₃
has at least one solution. Then, which of the following system(s) (in real variables) has (have) at least one
solution for each $\left[\begin{array}{ccc}b1\\b2\\b3\end{array}\right]$ ∊ S ?
(A) x + 2y + 3z = b₁, 4y + 5z = b₂ and x + 2y + 6z = b₃
(B) x + y + 3z = b₁, 5x + 2y + 6z = b₂ and -2x - y - 3z = b₃
(C) -x + 2y - 5z = b₁, 2x - 4y + 10z = b₂ and x - 2y + 5z = b₃
(D) x + 2y + 5z = b₁, 2x + 3z = b₂ and x + 4y - 5z = b₃

Answer 1

Answer:

Let S be the set of all column matrices $\left[\begin{array}{ccc}b1\\b2\\b3\end{array}\right]$ such that b₁, b₂, b₃ ∊ R and the system of equations (in real

variables)

-x + 2y + 5z = b₁

2x - 4y + 3z = b₂

x -2y + 2z = b₃

has at least one solution. Then, which of the following system(s) (in real variables) has (have) at least one

solution for each $\left[\begin{array}{ccc}b1\\b2\\b3\end{array}\right]$ ∊ S ?

(A) x + 2y + 3z = b₁, 4y + 5z = b₂ and x + 2y + 6z = b₃

(B) x + y + 3z = b₁, 5x + 2y + 6z = b₂ and -2x - y - 3z = b₃

(C) -x + 2y - 5z = b₁, 2x - 4y + 10z = b₂ and x - 2y + 5z = b₃

(D) x + 2y + 5z = b₁, 2x + 3z = b₂ and x + 4y - 5z = b₃