To determine which two equations form a system with infinite solutions, we need to find equations that are equivalent to each other. This often means manipulating the equations to check if they represent the same line (i.e., have the same slope and y-intercept).
Let's analyze each option:
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y = 5x - 4: This is in slope-intercept form (slope = 5, y-intercept = -4).
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y = -2x + 3: This is also in slope-intercept form (slope = -2, y-intercept = 3).
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y = 3x + 1: Again, slope-intercept form (slope = 3, y-intercept = 1).
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2y = 6x + 2: First, divide everything by 2: \( y = 3x + 1 \) (slope = 3, y-intercept = 1), which matches Option 3.
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y - 5x = 2: Rearranging gives \( y = 5x + 2 \) (slope = 5, y-intercept = 2).
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y - 47 + 1 = 0: This simplifies to \( y = 46 \) (horizontal line, slope = 0).
Now we can summarize relationships:
- Options 1 and 5 have the same slope (5) but different intercepts (-4 and 2), not infinite solutions.
- Options 2, 3, 4, and 6 do not match with others for infinite solutions.
Identified pairs:
- Option 3 (y = 3x + 1) and Option 4 (y = 3x + 1) represent the same line because they have the same slope and y-intercept.
Thus, only Option 3 and Option 4 can form a system with infinite solutions.
Final answer: Option 3 and Option 4.