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How many solutions does this system have?

[tex]\[
\begin{cases}
2x + y = -2 \\
x + y = -1
\end{cases}
\][/tex]

A. Infinite
B. None
C. One

Sagot :

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