Question
Which system of equations has no real solutions? (1 point)
Answers
Answered by
GPT-4o mini
To determine which system of equations has no real solutions, we generally look for scenarios where the lines represented by the equations are parallel and do not intersect.
A classic example of such a system would be two linear equations with the same slope but different y-intercepts. For instance:
1. \(y = 2x + 3\)
2. \(y = 2x - 5\)
In this case, both equations have the same slope (2), but different intercepts (3 and -5), meaning they will never intersect, thus having no real solutions.
Another common example is a system involving a quadratic and a linear equation where the quadratic does not intersect the line:
1. \(y = x^2 + 1\)
2. \(y = -2\)
In this case, the quadratic \(y = x^2 + 1\) is always above the line \(y = -2\) for all real \(x\), leading to no real solutions as well.
If you have a specific set of equations in mind, please provide them, and I can confirm whether they have real solutions or not.
A classic example of such a system would be two linear equations with the same slope but different y-intercepts. For instance:
1. \(y = 2x + 3\)
2. \(y = 2x - 5\)
In this case, both equations have the same slope (2), but different intercepts (3 and -5), meaning they will never intersect, thus having no real solutions.
Another common example is a system involving a quadratic and a linear equation where the quadratic does not intersect the line:
1. \(y = x^2 + 1\)
2. \(y = -2\)
In this case, the quadratic \(y = x^2 + 1\) is always above the line \(y = -2\) for all real \(x\), leading to no real solutions as well.
If you have a specific set of equations in mind, please provide them, and I can confirm whether they have real solutions or not.
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