Set of all points in 3 space?

Describe the set of all points (a, b, c) in 3-space with the property that the least squares approximate solution to the system of equations

$x – 2y = a$

$2x + y = b$

$3x + 2y = c$

is $x=1$ $y=-2$

I'm confused what this question is asking, it can't just mean input x and y into my system of equations, can it?

Solution 1:

Write your system in a matrix form $AX=Y$,

$$ \begin{bmatrix}1 & -2\\2 & 1 \\3 & 2 \end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}a \\b \\c\end{bmatrix}\,. $$

The technique to find the least square solution of the above system is to apply $A^T$ to both sides of the above equation. Doing that and substituting the given values $x=1$ and $y=-2$ yields

$$ \begin{bmatrix}1 & 2 & 3\\-2 & 1 & 2\end{bmatrix}\begin{bmatrix}1 & -2\\2 & 1 \\3 & 2 \end{bmatrix}\begin{bmatrix}1\\-2\end{bmatrix}= \begin{bmatrix}1 & 2 & 3\\-2 & 1 & 2\end{bmatrix}\begin{bmatrix}a \\b \\c\end{bmatrix}\,. $$

Work out the above and you will end up having two equations in $a,b,c$ which you need to solve. The solution describes the set of all $(a,b,c)\subset \mathbb{R^3} $