Answer
Consistent
Solution set: $\left\{\left(\dfrac{3}{2},1\right)\right\}$
Work Step by Step
We are given the system of equations:
$\begin{cases}
2x+3y=6\\
x-y=\frac{1}{2}
\end{cases}$
Write the augmented matrix:
$\begin{bmatrix}2&3&|&6\\1&-1&|&\frac{1}{2}\end{bmatrix}$
Perform row operations to bring the matrix to the reduced row echelon form:
$R_1=-r_2+r_1$
$\begin{bmatrix}1&4&|&\frac{11}{2}\\1&-1&|&\frac{1}{2}\end{bmatrix}$
$R_2=-r_1+r_2$
$\begin{bmatrix}1&4&|&\frac{11}{2}\\0&-5&|&-5\end{bmatrix}$
$R_2=-\dfrac{1}{5}r_2$
$\begin{bmatrix}1&4&|&\frac{11}{2}\\0&1&|&1\end{bmatrix}$
$R_1=-4r_2+r_1$
$\begin{bmatrix}1&0&|&\frac{3}{2}\\0&1&|&1\end{bmatrix}$
Write the corresponding system of equations:
$\begin{cases}
1x+0y=\frac{3}{2}\\
0x+1y=1
\end{cases}$
$\begin{cases}
x=\frac{3}{2}\\
y=1
\end{cases}$
The system is consistent. The solution set is:
$\left\{\left(\dfrac{3}{2},1\right)\right\}$
