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Which system of equations has no real solutions?(1 point)

Responses

y= x2 and y= x +1
y equals x squared and y equals x plus 1

y = x2+1 and y =−x2 −1
y equals x squared plus 1 and y equals negative x squared minus 1

y=3x+1 and y=2x+2
y equals 3 x plus 1 and y equals 2 x plus 2

y = −x2+2 and y=x2

User Icon for Bot GPT 4-o mini Bot GPT 4-o mini answered
25 days ago

To determine which system of equations has no real solutions, we can look at each pair of equations and analyze their graphs.

1. **y = x² and y = x + 1**: The first equation is a parabola opening upwards, while the second is a line with a positive slope. They intersect at two points, so they have real solutions.

2. **y = x² + 1 and y = -x² - 1**: The first equation is a parabola opening upwards, and the second is a downward opening parabola. The lowest point (vertex) of y = x² + 1 is at (0, 1), while the highest point of y = -x² - 1 is at (0, -1). Since the first parabola (y = x² + 1) is always above the line y = -1 and they do not intersect, this system has no real solutions.

3. **y = 3x + 1 and y = 2x + 2**: Both of these are linear equations. They have different slopes (3 and 2), so they intersect at one point and thus have real solutions.

4. **y = -x² + 2 and y = x²**: The first equation is a downward-opening parabola, while the second is an upward-opening parabola. They intersect at two points (when both equal to each other). Therefore, this system has real solutions.

Based on the analysis, the system that has **no real solutions** is:

**y = x² + 1 and y = -x² - 1.**