Answer
$(2,-1)$
Work Step by Step
To solve the system $\begin{cases}x+y=1 \\ x-2y=4 \\ \end{cases},$ we perform elementary row operations on the corresponding augmented matrix to obtain an equivalent matrix with $1s$ along the main diagonal.
The corresponding augmented matrix is
$$\left[
\begin{array}{cc|c}
1 & 1 & 1 \\
1 & -2&4\\
\end{array}
\right].$$
We replace Row_2 with Row_2 minus Row_1 to obtain the equivalent matrix
$$\left[
\begin{array}{cc|c}
1 & 1 & 1 \\
0 & -3 & 3\\
\end{array}
\right].$$
Next, we multiply Row_2 by $-\dfrac{1}{3}$ to obtain the equaivalent matrix
$$\left[
\begin{array}{cc|c}
1 & 1 & 1 \\
0 & 1&-1\\
\end{array}
\right].$$
Now, we form the system of equations corresponding to this matrix
$$\begin{cases}x+y=1 \\ y=-1 \\ \end{cases}.$$
We see $y=-1$, and to find our value for $x$, we plug $y=-1$ into the first equation and solve it for $x$.
Thus
$$x+(-1)=1$$
gives
$$x=2$$.
Thus we have exactly one solution, which is $(2,-1).$