Let's analyze the given system of equations step by step to determine the nature of the system. The system of equations provided is:
[tex]\[
\begin{cases}
y = -2x + 5 \\
2x + y = -5
\end{cases}
\][/tex]
### Step 1: Understand the Equations
First, we rewrite both equations:
1. [tex]\( y = -2x + 5 \)[/tex]
2. [tex]\( 2x + y = -5 \)[/tex]
### Step 2: Substitute [tex]\( y \)[/tex] from the First Equation into the Second Equation
We substitute the expression for [tex]\( y \)[/tex] from the first equation into the second equation:
[tex]\[
2x + (-2x + 5) = -5
\][/tex]
Simplify the equation:
[tex]\[
2x - 2x + 5 = -5 \\
0 + 5 = -5 \\
5 = -5
\][/tex]
### Step 3: Analyze the Result
This simplifies to a false statement [tex]\( 5 = -5 \)[/tex]. Because this is a contradiction, it implies that there is no set of values for [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that can satisfy both equations simultaneously.
### Step 4: Draw a Conclusion
Since we have arrived at a contradiction, this tells us that there is no solution to this system of equations. Therefore, the system is inconsistent.
### Final Statement
The correct statement about the system is:
The system is inconsistent because it has no solution.
