Problem 26 - Algebra and Trigonometry

Question

Find an equation of the line that satisfies the given conditions. Through \((4,5) ; \quad\) parallel to the \(y\) -axis

Step-by-Step Solution

Answer

The equation of the line is \(x = 4\).

Step 1: Understanding the problem

We need to find the equation of a line that passes through the point \((4,5)\) and is parallel to the \(y\)-axis.

Step 2: Identifying characteristics of the line

A line that is parallel to the \(y\)-axis is a vertical line. Vertical lines have the same \(x\)-coordinate for all their points. Hence, the equation of a vertical line is of the form \(x = a\).

Step 3: Determining the line's equation

Since the line passes through the point \((4,5)\), the \(x\)-coordinate of all points on the line is \(4\). Therefore, the equation of the line is \(x = 4\).

Key Concepts

Vertical Line

A vertical line in coordinate geometry is unique because it defies the standard equation format of y = mx + b, which depicts a line with both slope and y-intercept. Instead, vertical lines have an equation in the form of x = a constant. This equation shows that regardless of what the y-coordinate is, the x-coordinate always remains the same.

Characteristics of vertical lines include being parallel to the y-axis, and not having a defined slope since their change in y is undefined. Because the x-coordinates remain consistent, any point (x, y) on a vertical line will share the same x value.

Vertical lines are also infinite along the y-direction and their graph is simple: it is a straight line that runs up and down the coordinate plane without bending or angling.

Parallel to Y-axis

When a line is said to be parallel to the y-axis, it means the line runs vertically along the grid without deviation. This orientation means that for every point on the line, the x-coordinate remains constant while the y-coordinate can take any value.

To understand it better, consider a conventional grid. The y-axis itself is a vertical line, extending north and south on the grid. A vertical line parallel to this axis will not intersect it, showing consistency with the characteristic constant x-coordinate.

Such lines are crucial in dividing spaces or compartments in coordinate planes, and knowing their properties is essential for grasping more complex mathematical concepts like geometric transformations and reflections.

Coordinate Geometry

Coordinate geometry, often known as analytic geometry, is the study of geometric figures using the coordinate plane. It marries algebra and geometry, allowing us to derive equations and relationships that represent shapes and movements in space.

Using a set of axes, typically the x and y-axes, coordinate geometry helps describe positions and distances. Each point on this plane is identified by an ordered pair of numbers (x, y), telling us precisely where it is located.

Fundamental concepts involve plotting basic shapes like lines, ellipses, and circles. For example, the equation x = 4 represents a vertical line that remains parallel to the y-axis across all y-values, showing how coordinate geometry helps visualize mathematical ideas. This makes it an invaluable tool for solving real-world problems and understanding the spatial world around us.

• Empowers understanding of distance, midpoint, and division of lines.
• Facilitates understanding geometric transformations like translations and rotations.
• Used extensively in fields such as physics to represent trajectories and forces.