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Question:

Which statement about the following system is correct?

[tex]\[
\begin{array}{l}
y = -2x + 5 \\
2x + y = -5
\end{array}
\][/tex]

A. The system is consistent because it has no solution.
B. The system is inconsistent because it has no solution.
C. The system is inconsistent because the two equations actually represent the same line.
D. The system is consistent because it has a solution.



Answer :

GinnyAnswer
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.

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