Finding solutions to matrices (should be easy)

[Ush]

Homework Statement

Alvin is informed that the homogeneous system of equations AX = 0 has a
one parameter family of solutions given by

X = t[4 3 0]^T

By trial and error, he has found that

X = [-1 5 7]^T

is a solution to the inhomogenous problem

AX = B

where

B = [3 -2 -5]^T

He suspects there are more solutions

a) Is there a solution to AX=B of the form [1 a b]^T? If so, what are a and b?
b) Is there a solution to AX=B of the form [c d 1]^T? If so, what are c and d?

Homework Equations

AX = B
X = BA^-1

The Attempt at a Solution

I'm more used to simple matrices such as the following

I just don't know how to apply this concept to this question,

my test is tomorrow, any help would be appreciated =)

 

[Mark44]

Homework Statement

Alvin is informed that the homogeneous system of equations AX = 0 has a
one parameter family of solutions given by

X = t[4 3 0]^T

By trial and error, he has found that

X = [-1 5 7]^T

is a solution to the inhomogenous problem

AX = B

where

B = [3 -2 -5]^T

He suspects there are more solutions

a) Is there a solution to AX=B of the form [1 a b]^T? If so, what are a and b?
b) Is there a solution to AX=B of the form [c d 1]^T? If so, what are c and d?

Homework Equations

AX = B
X = BA^-1

The second equation above is true if and only if A is invertible.

The Attempt at a Solution

From the given information, we can say that
$$A t\begin{bmatrix}4\\3\\0\end{bmatrix} + A \begin{bmatrix}-1\\5\\7\end{bmatrix} = \begin{bmatrix}0\\0\\0\end{bmatrix} + \begin{bmatrix}3\\-2\\-5\end{bmatrix}$$
for any value of t.

For a), Is there some particular value of t for which the two vectors on the right add up to <1, a, b>^T?
The b part is similar.