Question
We are told that a circle has the following 3 points lie on its circumference: $A=(-1,-3), B=(0,-1)$ and $ C=(-1,1).$
I have this polynomial system. In order to complete the problem, we must solve the system of 3 linear equations:
$-D-3E+F=-10,$ $-E+F=-1 $ and $-D+E+F=-2$
Which came from replacing each point in the general equation of the circumference.
My Attempt
Every time I complete the system I end up with different values for $D, E$ and $F. $
And the end values for the centre is $(-5/2,-1)$ and radius: $5/2$
How do I resolve this contradiction?
We can write this out as a matrix and apply Gaussian elimination (row reducing the matrix into a simpler form to deal with). This is because we have a system of linear equations where we have 3 equations and 3 unknown variables to find.
We start with the matrix where each row represents one of the three equations in the original question and then perform elementary row operations to get this into a simpler form.
$\begin{pmatrix}-1 & -3 & 1 & -10\\\ 0 & -1 & 1 & -1\\\ -1 & 1 & 1 & -2 \end{pmatrix}$ $\rightarrow$ $\begin{pmatrix}1 & 3 & -1 & 10\\\ 0 & -1 & 1 & -1\\\ 0 & 4 & 0 & 8 \end{pmatrix}$$\rightarrow$ $\begin{pmatrix}1 & 0 & 2 & 7\\\ 0 & 1 & -1 & 1\\\ 0 & 0 & 1 & 1 \end{pmatrix}$$\rightarrow$ $\begin{pmatrix}1 & 0 & 2 & 7\\\ 0 & 1 & 0 & 2\\\ 0 & 0 & 1 & 1 \end{pmatrix}$
We could continue, but it is clear what the solution is from here.
We see that $F = 1$ from the final row. We see that $E=2$ from the second row. And the first row tells us that $D+2F=7$ which tells us $D=5$ by substitution for $F$ using the value that we have just found.
Therefore, we are done. We have found that $D=5,E=2,F=1$. We can double check this by substitution back into the original equations and we see that this returns the correct answer in all 3 (showing that we have found the correct solution).