To determine which two equations form a system with no solutions, we need to identify equations that have the same slope but different y-intercepts. This will happen when the lines represented by the equations are parallel.
Let's rewrite each equation in slope-intercept form (y = mx + b) to easily identify the slope (m) and y-intercept (b).
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\( y = 5x - 4 \)
- Slope: \( m = 5 \)
- Y-intercept: \( b = -4 \)
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\( y = -2x + 3 \)
- Slope: \( m = -2 \)
- Y-intercept: \( b = 3 \)
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\( y = 3x + 1 \)
- Slope: \( m = 3 \)
- Y-intercept: \( b = 1 \)
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\( 2y = 6x + 2 \)
- Divide by 2:
- \( y = 3x + 1 \)
- Slope: \( m = 3 \)
- Y-intercept: \( b = 1 \)
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\( y - 5x = 2 \)
- Rearranging gives:
- \( y = 5x + 2 \)
- Slope: \( m = 5 \)
- Y-intercept: \( b = 2 \)
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\( y - 4x + 1 = 0 \)
- Rearranging gives:
- \( y = 4x - 1 \)
- Slope: \( m = 4 \)
- Y-intercept: \( b = -1 \)
Now let's compare the slopes:
- \( y = 5x - 4 \) and \( y = 5x + 2 \) both have a slope of 5 (same slope) but different y-intercepts (-4 and 2). Therefore, they are parallel lines and form a system with no solutions.
So, the two equations that form a system with no solutions are:
- \( y = 5x - 4 \) (Equation 1)
- \( y = 5x + 2 \) (Equation 5)