Answer
No solution or $\varnothing$.
Work Step by Step
The given system of equations is
$4x-2y=5$
$-2x+y=6$
The augmented matrix is
$\Rightarrow \left[\begin{array}{cc|c}
4 & -2 & 5\\
-2 & 1 & 6
\end{array}\right]$
Perform $R_1\rightarrow \frac{R_1}{4}$.
$\Rightarrow \left[\begin{array}{cc|c}
4/4 & -2/4 & 5/4 \\
-2 & 1 & 6
\end{array}\right]$
Simplify.
$\Rightarrow \left[\begin{array}{cc|c}
1 & -1/2& 5/4 \\
-2 & 1 & 6
\end{array}\right]$
Perform $R_2\rightarrow R_2+2\times R_1$.
$\Rightarrow \left[\begin{array}{cc|c}
1 & -1/2 & 5/4\\
-2+2\times 1 & 1-2\times (1/2) & 6+2\times (5/4)
\end{array}\right]$
Simplify.
$\Rightarrow \left[\begin{array}{cc|c}
1 & -1/2 & 5/4\\
0 & 0 & 17/2
\end{array}\right]$
Use back substitution to solve the linear system.
$\Rightarrow (1)x+(-1/2)y=5/4$
and
$\Rightarrow (0)x+(0)y=17/2$
In the second equation no values of $x$ and $y$ satisfy.
Hence, the system is inconsistent and has no solution.
The solution set is $\varnothing$.
