Understanding the Equation x = 5 on the xy-Plane

The question asks us to determine the total number of points that satisfy the equation \(x = 5\) when plotted on the \(xy\)-plane.

The \(xy\)-plane is a two-dimensional coordinate system where each point is represented by an ordered pair \((x, y)\). The first value, \(x\), represents the horizontal position (horizontal axis or x-axis), and the second value, \(y\), represents the vertical position (vertical axis or y-axis).

Analyzing the Equation x = 5

The given equation is \(x = 5\). This equation specifically sets a condition on the \(x\)-coordinate only. It states that for any point to satisfy this equation, its \(x\)-coordinate must be exactly 5.

Crucially, the equation places no restriction whatsoever on the \(y\)-coordinate. The \(y\)-coordinate can take any real value.

Identifying Points Satisfying x = 5

Let's consider some examples of points that fit the equation \(x = 5\):

• If \(y=0\), the point is \((5, 0)\).
• If \(y=1\), the point is \((5, 1)\).
• If \(y=-3\), the point is \((5, -3)\).
• If \(y=100\), the point is \((5, 100)\).
• If \(y = \pi\) (approximately 3.14159...), the point is \((5, \pi)\).
• If \(y = -50.5\), the point is \((5, -50.5)\).

As you can see, for every possible real number we choose for \(y\), we get a unique point on the \(xy\)-plane where the \(x\)-coordinate is 5. Geometrically, this represents a vertical line passing through \(x=5\) on the x-axis.

Conclusion on the Number of Points

Since there are infinitely many real numbers that the \(y\)-coordinate can be, there must be infinitely many points \((5, y)\) that satisfy the equation \(x = 5\). Therefore, the equation \(x = 5\) represents infinitely many points on the \(xy\)-plane.