Let's analyze each system of equations to classify them:
A. \[ \begin{cases} x + 5y = -2 \ x + 5y = 4 \end{cases} \] These two equations represent parallel lines (same left side but different right sides), which means they have no point of intersection. Classification: 3. Inconsistent
B. \[ \begin{cases} y = 3x + 4 \ -2x + y = 4 \end{cases} \] First, let's rewrite the second equation in slope-intercept form: \[ y = 2x + 4 \] Now we have: \[ \begin{cases} y = 3x + 4 \ y = 2x + 4 \end{cases} \] These two equations have different slopes (3 and 2), so they are not parallel and will intersect at one point. Classification: 1. Consistent independent
C. \[ \begin{cases} 3x + y = 4 \ -6x - 2y = -8 \end{cases} \] Let's simplify the second equation: \[ -6x - 2y = -8 \implies 3x + y = 4 \quad \text{(divide the entire equation by -2)} \] Thus, both equations are identical. When two equations are the same, they have infinitely many solutions (they represent the same line). Classification: 2. Consistent dependent
Now we can summarize the classifications:
- A: 3. Inconsistent
- B: 1. Consistent independent
- C: 2. Consistent dependent
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