Usually, the question can contain long text before it. You should give it also.
No Solutions
One Solution
Infinitely Many Solutions
Sure! Here’s the question without answer options: “Determine the number of solutions for the following system of equations given the conditions that there can be No Solutions, One Solution, or Infinitely Many Solutions.”
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To determine the number of solutions a given equation or system of equations has, we can analyze it as follows:
1. No Solutions: This occurs when the equations contradict each other, meaning there is no set of values for the variables that can satisfy all equations simultaneously. An example is the equations (x + 2 = 5) and (x + 2 = 6); these create parallel lines that never intersect.
2. One Solution: This occurs when there is exactly one set of values for the variables that satisfies the equations. In a system of two linear equations, this happens when the lines intersect at a single point. For example, the equations (x + y = 2) and (x – y = 0) will intersect at one unique point.
3. Infinitely Many Solutions: This is the case when the equations represent the same line, meaning any solution for one equation is also a solution for the other. For example, the equations (2x + 4y = 8) and (x + 2y = 4) are equivalent, leading to infinitely many solutions.
To summarize, the solution set depends on the relationship between the equations involved: conflicting, unique intersection, or identical representations. If you have a specific equation or system you’d like to discuss, feel free to share!