Question about solution to matrix equation

Question

In the question above, I am unsure how the very first step of the solution came about. How does fact that R is in reduced row echelon form help deduce the value of d?

Answer 1

Recall that for the general solution of a linear system is given by

$$x=x_p+x_h$$

with

• $x_p$ particular solution such that $$Ax_p=b$$

• $x_h$ homogeneous solution such that $$Ax_h=0$$

ans since R has rank $1$ it is equal to

$$R=\begin{bmatrix}1&0&0\\0&0&0\\0&0&0\end{bmatrix}$$

What can we conclude now?