Problem 9 - Algebra and Trigonometry

Question

Sketch the region given by the set. \(\\{(x, y) | x=3\\}\)

Step-by-Step Solution

Answer

The region is a vertical line at \( x = 3 \) extending infinitely in the \( y \)-direction.

Step 1: Identify the Condition

The given set is \( \{(x, y) | x=3\} \). This means that we are looking at all points \((x, y)\) where the x-coordinate is fixed at 3.

Step 2: Interpret the Condition

Since \( x = 3 \), the set includes every possible \( y \)-coordinate, meaning we have a vertical line where every point has the form \((3, y)\).

Step 3: Determine the Range of Values

For the line \( x = 3 \), the \( y \)-values are unrestricted, so they span from negative infinity to positive infinity. This line extends infinitely in the vertical direction.

Step 4: Sketch the Region

To sketch the region determined by \( x = 3 \), draw a vertical line on the coordinate plane that intersects the x-axis at \( x = 3 \). Ensure the line is vertical since all points on this line satisfy the condition \( x = 3 \).

Key Concepts

Vertical Line

A vertical line, in coordinate geometry, is a straight line where all the points share the same x-coordinate. This means that if you know the x-coordinate of any point on a vertical line, you can deduce that it is the same for all points on that line. In this specific exercise, the equation given, \(x = 3\), describes such a vertical line.

Understanding vertical lines is crucial because they are one of the simplest forms of linear equations. They are unique in that:

• Their slope is undefined: Unlike other lines, which have numerical slope values, vertical lines go straight up and down, making their slope undefined.
• They do not cross the y-axis: Instead, they are parallel to it and intersect the x-axis at their designated x-value.

Drawing a vertical line on a graph involves plotting a series of points where the x-coordinate is the same and connecting them. This line will extend infinitely in the vertical direction as indicated in the equation.

Graphing

Graphing lines, such as the vertical line \(x = 3\), is a fundamental part of coordinate geometry. It allows you to visualize equations and understand the relationship between x and y coordinates. To graph the line represented by \(x = 3\):

• Find the position on the x-axis corresponding to the value 3.
• From this point, draw a straight line that extends upward and downward indefinitely, ensuring that it remains parallel to the y-axis.

Graphing helps in providing a visual representation, which can make understanding mathematical relationships easier. For students, this visual aid is a powerful tool because it transforms numbers and equations into something tangible.

Therefore, whenever you are given an equation like \(x = 3\), it's important to remember that graphing it as a vertical line reinforces the concept that y can take any value.

Infinite Set

An infinite set in mathematics is a set with no bounds. It keeps going on without ending. For vertical lines, like \(x = 3\), the concept of an infinite set comes into play through the y-coordinates, which can take any value from negative infinity to positive infinity.In this context:

• The x-coordinate stays constant at 3, as indicated by the equation.
• The set of y-values is infinite, signifying that the line extends upwards and downwards indefinitely without any bounds.

Visualizing an infinite set might seem challenging, but recognizing that it simply implies never-ending values makes it easier to grasp. Vertical lines inherently possess this infinite nature in the context of their y-values, which aligns perfectly with the mathematical definition of an infinite set.