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Bot GPT 4-o mini To determine which system of equations has no real solutions, we need to analyze each pair of equations to see if they intersect or not.
1. **y = x² + 1 and y = -x² - 1**
- The first equation (y = x² + 1) is a parabola that opens upwards with a vertex at (0, 1).
- The second equation (y = -x² - 1) is a parabola that opens downwards with a vertex at (0, -1).
- Since the vertex of the first parabola is above the x-axis, and the vertex of the second parabola is below the x-axis, these two curves will not intersect.
**No real solutions.**
2. **y = 3x + 1 and y = 2x + 2**
- These are both linear equations. The first equation has a slope of 3 and the second one has a slope of 2. Since their slopes are different, they will intersect at exactly one point.
**Has real solutions.**
3. **y = -x² + 2 and y = x²**
- Both are parabolas. The first one opens downward and the second one opens upward. Setting -x² + 2 = x² leads to -2x² + 2 = 0, which means x² = 1, thus x = ±1. So, they intersect at two points.
**Has real solutions.**
4. **y = x² and y = x + 1**
- The first equation is a parabola opening upwards, while the second equation is a line. Setting x² = x + 1 leads to x² - x - 1 = 0, which has real solutions (by the quadratic formula: x = (1 ± √5) / 2). So they intersect at two points.
**Has real solutions.**
In summary, the only system of equations without real solutions is:
**y = x² + 1 and y = -x² - 1.**