Question

The system of equations \(x = 2\) and \(x = 3\) has:

• A. unique solution (2, 3)
• B. two solutions (2, 0) and (3, 0)
• C. no solution
• D. infinitely many solutions

Hint:

Graphically, the solution is the point of intersection. Since a single point cannot have an x-value of both 2 and 3 at the same time, it's impossible for these lines to meet.

Answer 1

The Correct Option is C

This problem involves the analysis of a system of equations, specifically two linear equations in the form:

• x = 2
• x = 3

Let's analyze these equations:

1. The equation x = 2 represents a vertical line on the Cartesian plane where every point on this line has an x-coordinate equal to 2.
2. Similarly, the equation x = 3 represents another vertical line on the Cartesian plane where every point on this line has an x-coordinate equal to 3.

Now, let's examine the interaction of these two lines:

• Vertical lines are parallel if they have different x-coordinates, which is the case here because 2 is not equal to 3.
• Since these lines are parallel, they do not intersect at any point.

In the context of finding solutions to a system of equations, a solution corresponds to a point of intersection. Therefore:

• No Intersection: Since the lines do not intersect, there is no point (x, y) that satisfies both equations simultaneously.

Consequently, the given system of equations has no solution.

Let's confirm the correctness of the options:

• Unique solution (2, 3): Incorrect. A unique solution would require intersection at a single point, which does not occur here.
• Two solutions (2, 0) and (3, 0): Incorrect. These points do not satisfy both equations simultaneously, as the equations imply intersecting lines, which is not the case.
• No solution: Correct. Parallel vertical lines imply no intersection, thus, no solution exists.
• Infinitely many solutions: Incorrect. Infinitely many solutions would imply the lines overlap, which they do not.

The correct answer is: No Solution.