To determine the number of solutions for the given system of equations:
[tex]\[
\left\{\begin{array}{r}
2x + y = -2 \\
x + y = -1
\end{array}\right.
\][/tex]
we can proceed as follows:
1. Write down the system of equations:
[tex]\[
\begin{cases}
2x + y = -2 \quad \text{(1)} \\
x + y = -1 \quad \text{(2)}
\end{cases}
\][/tex]
2. Represent the system in matrix form [tex]\( \mathbf{A} \mathbf{x} = \mathbf{b} \)[/tex]:
[tex]\[
\mathbf{A} = \begin{pmatrix}
2 & 1 \\
1 & 1
\end{pmatrix}, \quad
\mathbf{x} = \begin{pmatrix}
x \\
y
\end{pmatrix}, \quad
\mathbf{b} = \begin{pmatrix}
-2 \\
-1
\end{pmatrix}
\][/tex]
3. Calculate the determinant of the coefficient matrix [tex]\( \mathbf{A} \)[/tex]:
The determinant [tex]\( \Delta \)[/tex] of matrix [tex]\( \mathbf{A} \)[/tex] can be determined as follows:
[tex]\[
\Delta = \begin{vmatrix}
2 & 1 \\
1 & 1
\end{vmatrix} = (2 \cdot 1) - (1 \cdot 1) = 2 - 1 = 1
\][/tex]
4. Analyze the determinant:
Since the determinant [tex]\( \Delta \)[/tex] is [tex]\( 1 \)[/tex], which is non-zero, it implies that the coefficient matrix [tex]\( \mathbf{A} \)[/tex] is invertible. This means the system of equations has a unique solution.
By this analysis, we conclude that the given system of linear equations has exactly one solution.
Thus, the correct answer is:
- One
