To determine the number of solutions for the given system of equations, we should analyze the slopes and intercepts of the lines represented by each equation.
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y = 5x + 7
- Slope = 5
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3y - 15x = 21
- Rewriting gives: y = 5x + 7
- Slope = 5 (same as first equation)
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y = -13x - 4
- Slope = -13 (different slope)
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y = 3x + 6
- Slope = 3 (different slope)
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y = 5x - 1
- Slope = 5 (same slope as 1st and 2nd equations)
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y = 5x + 2
- Slope = 5 (same slope as 1st, 2nd, and 5th equations)
Now we categorize based on the slopes:
- Equations 1, 2, 5, and 6 have the same slope (5), meaning they are parallel and hence do not intersect. They do not share any solutions.
- Equations 3 and 4 have different slopes (-13 and 3 respectively). They will intersect with each other and with the equations with slope 5 at two distinct points (since they are not parallel to each other).
Conclusion
Therefore, we can conclude:
- The equations with the same slope (y = 5x + 7; 3y - 15x = 21; y = 5x - 1; y = 5x + 2) have zero solutions among themselves.
- Thus, you have two distinct lines (which will intersect) represented by equations with different slopes.
Final Matching
- Equations with zero solutions: y = 5x + 7; 3y - 15x = 21; y = 5x - 1; y = 5x + 2
- Equations that intersect (at least one solution): y = -13x - 4; y = 3x + 6
This gives you an overall understanding of the number of solutions for the various equations presented.