To determine which system describes an infinite number of solutions, we look for two equations that represent the same line. This occurs when the equations are proportional.
Let's analyze each pair:
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First pair: \[ -8x - 5y = 6 \quad \text{(1)} \] \[ -8x - 5y = 12 \quad \text{(2)} \]
- These two equations have the same left-hand side but different right-hand sides. Therefore, they represent parallel lines and do not have any solutions, let alone an infinite number of solutions.
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Second pair: \[ 8x + 5y = 6 \quad \text{(1)} \] \[ 5x + 8y = 6 \quad \text{(2)} \]
- These equations are not multiples of each other, so they represent two different lines that will intersect at one point. Therefore, they have a unique solution, not infinite.
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Third pair: \[ -10x + 4y = 6 \quad \text{(1)} \] \[ -5x + 2y = 3 \quad \text{(2)} \]
- To see if they are proportional, we can rewrite the second equation by multiplying it by 2: \[ -10x + 4y = 6 \]
- This gives us the same equation as the first. Thus, these two equations represent the same line, meaning they have an infinite number of solutions.
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Fourth pair: \[ -10x - 4y = -6 \quad \text{(1)} \] \[ 10x + 4y = 2 \quad \text{(2)} \]
- These equations do not represent the same line; they also do not satisfy proportionality.
From the analysis, the system that describes an infinite number of solutions is:
\[ -10x + 4y = 6 \quad \text{and} \quad -5x + 2y = 3 \]