To find the equations that form a system with infinite solutions, we need to identify equations that are equivalent, meaning they have the same slope (m) and y-intercept (b). Let's first convert equations #4, #5, and #6 into the y = mx + b format.
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Equation #4: \(2y = 6x + 2\)
Divide by 2: \[ y = 3x + 1 \]
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Equation #5: \(y - 5x = 2\)
Add \(5x\) to both sides: \[ y = 5x + 2 \]
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Equation #6: \(y - 4x + 1 = 0\)
Rearranging gives us: \[ y = 4x - 1 \]
Now we have the following equations in y = mx + b format:
- \(y = 5x - 4\) (from Option 1)
- \(y = -2x + 3\) (from Option 2)
- \(y = 3x + 1\) (from option #4 converted)
- \(y = 5x + 2\) (from option #5 converted)
- \(y = 4x - 1\) (from option #6 converted)
Next, we need to identify pairs of equations that have the same slope (m) and the same y-intercept (b):
- Equations 1 and 4: \(y = 5x - 4\) and \(y = 5x + 2\) (different y-intercepts)
- Equations 1 and 5: \(y = 5x - 4\) and \(y = 5x + 2\) (different y-intercepts)
- Equations 4 and 3: \(y = 3x + 1\) and \(y = 3x + 1\) (same slope and intercept – infinite solutions)
So the only pair of equations from this list with the same slope and the same y-intercept is:
- Equations 3 and 4: (y = 3x + 1 and y = 3x + 1) (both equations yield infinite solutions)
Thus, the answer is the pair of equations in Option 3 and Option 4.