To determine the number of solutions for the given system of linear equations, let's analyze the system step by step.
The system of equations is:
[tex]\[
\left\{
\begin{array}{r}
2x + y = -2 \\
x + y = -1
\end{array}
\right.
\][/tex]
### Step 1: Compare the Equations
We have two equations with two variables (x and y). These are:
1. [tex]\(2x + y = -2\)[/tex]
2. [tex]\(x + y = -1\)[/tex]
### Step 2: Eliminate One Variable
To solve the system, we can try to eliminate one of the variables by manipulating the equations. Let's subtract the second equation from the first:
[tex]\[
(2x + y) - (x + y) = -2 - (-1)
\][/tex]
Simplifying the left side:
[tex]\[
2x + y - x - y = -2 + 1
\][/tex]
[tex]\[
x = -1
\][/tex]
So we've found that [tex]\(x = -1\)[/tex].
### Step 3: Solve for the Other Variable
Next, let's substitute [tex]\(x = -1\)[/tex] back into one of the original equations to solve for [tex]\(y\)[/tex]. We can use the second equation:
[tex]\[
x + y = -1
\][/tex]
[tex]\[
-1 + y = -1
\][/tex]
[tex]\[
y = 0
\][/tex]
### Step 4: Check Consistency
Substitute [tex]\(x = -1\)[/tex] and [tex]\(y = 0\)[/tex] back into the first equation to check for consistency:
[tex]\[
2(-1) + 0 = -2
\][/tex]
[tex]\[
-2 = -2
\][/tex]
The values satisfy both equations, indicating they are consistent.
### Conclusion
Since we have found one unique solution [tex]\((x, y) = (-1, 0)\)[/tex], and the equations are consistent, we determine that:
[tex]\[
\text{The system has one solution.}
\][/tex]
Therefore, the number of solutions is:
One
