Which statement about the following system is correct? \( \begin{array}{l}y=-2 x+5 \\ 2 x+y=-5 \\ \text { The system is inconsistent because the two equations actually represent the same line } \\ \text { The system is inconsistent because it has no solution. } \\ \text { The system is consistent because it has a solution. } \\ \text { The system is consistent because it has no solution. }\end{array} \). The

Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}y=-2x+5\\2x+y=-5\end{array}\right.\) - step1: Substitute the value of \(y:\) \(2x-2x+5=-5\) - step2: Simplify: \(5=-5\) - step3: Calculate: \(\left\{ \begin{array}{l}x \in \varnothing \\y \in \varnothing \end{array}\right.\) - step4: Rewrite: \((x, y) \in \varnothing\) To analyze the given system of equations: 1. The first equation is \( y = -2x + 5 \). 2. The second equation is \( 2x + y = -5 \). We can rewrite the second equation in slope-intercept form to compare it with the first equation: \[ y = -2x - 5 \] Now we have: - First equation: \( y = -2x + 5 \) - Second equation: \( y = -2x - 5 \) ### Step 1: Compare the slopes and intercepts Both equations have the same slope of \(-2\), but different y-intercepts (5 and -5). This means that the lines are parallel and will never intersect. ### Step 2: Determine the consistency of the system Since the two lines are parallel and do not intersect, the system of equations has no solution. ### Conclusion The correct statement about the system is: - **The system is inconsistent because it has no solution.** Thus, the answer is: **The system is inconsistent because it has no solution.**