Answer
Our solution set is $\{(x,y): 9x-3y=6\}$.
Work Step by Step
To solve the system $\begin{cases}9x-3y=6 \\ -18x+6y=-12 \\ \end{cases},$ we perform elementary row operations on the corresponding augmented matrix to obtain an equivalent matrix with $1s$ along the main diagonal (if possible).
The corresponding augmented matrix is
$$\left[
\begin{array}{cc|c}
9 & -3 & 6 \\
-18 & 6 & -12\\
\end{array}
\right].$$
We replace Row_2 with Row_2+2*Row_1 to obtain the equivalent matrix
$$\left[
\begin{array}{cc|c}
9 & -3 & 6 \\
0 & 0 & 0\\
\end{array}
\right].$$
Now, we see every entry in Row_2 is zero. This means the system of equations corresponding to this matrix is consistent and has infinitely many solutions.
To see this, we form the system of equations corresponding to this matrix:
$$\begin{cases}9x-3y=6 \\ 0=0 \\ \end{cases}.$$
We see $0=0$ is true for all values of $x$ (or, equivalently, all values of $y$).
So the ordered pairs $(x,y)$ that satisfy this system of equations are the ordered pairs $(x,y)$ that satisfy the equation $9x-3y=6$.
Hence every ordered pair in the set $\{(x,y): 9x-3y=6\}$ is a solution to our system.
