Problem 20 - Precalculus

Question

Write an equation of: (a) a vertical line passing through the given point; (b) a horizontal line passing through the given point. $$(5,8)$$

Step-by-Step Solution

Answer

(a) Vertical line: \(x = 5\); (b) Horizontal line: \(y = 8\).

Step 1: Understand Vertical Line Equation

A vertical line has an undefined slope and its equation is of the form \( x = a \), where \( a \) is the x-coordinate of every point on the line. Since the line is vertical, the y-coordinate can be anything, but the x-coordinate is constant.

Step 2: Identify x-coordinate from the Point

In this exercise, the point given is \((5, 8)\). For the vertical line passing through this point, focus on the x-coordinate, which is 5.

Step 3: Write Equation for Vertical Line

The equation of the vertical line passing through the point \((5, 8)\) is \( x = 5 \). This line represents all points with x-coordinate equal to 5.

Step 4: Understand Horizontal Line Equation

A horizontal line has a slope of 0 and its equation is of the form \( y = b \), where \( b \) is the y-coordinate of every point on the line. Since the line is horizontal, the x-coordinate can be anything, but the y-coordinate is constant.

Step 5: Identify y-coordinate from the Point

Using the same point \((5, 8)\), for the horizontal line passing through this point, focus on the y-coordinate, which is 8.

Step 6: Write Equation for Horizontal Line

The equation of the horizontal line passing through the point \((5, 8)\) is \( y = 8 \). This line represents all points with y-coordinate equal to 8.

Key Concepts

Vertical Line Equation

In coordinate geometry, vertical lines play a critical role. The equation of a vertical line is straightforward and follows a consistent pattern. A vertical line goes straight up and down and does not tilt like other lines. Therefore, it has what we call an "undefined" slope. The reason the slope is undefined is that a vertical line cannot be measured in terms of how much it 'rises' over a 'run' between two points. The run is zero because it does not move horizontally!

The equation of any vertical line can be written in the form:

• \( x = a \)

This equation tells us that no matter what the value of \( y \) is, \( x \) will always be equal to a constant \( a \). For a vertical line passing through the point \((5, 8)\), the equation is \( x = 5 \). This implies that all points on this line will have an x-coordinate of 5, but their y-coordinates could be any value.

Horizontal Line Equation

Horizontal lines are another fundamental type of line in coordinate geometry. Unlike vertical lines, horizontal lines do not climb or fall; they maintain a steady altitude. This is why a horizontal line's slope is zero. The idea is that for every step to the right (or left), there is no change in vertical position.

The equation of a horizontal line is simple and takes the following form:

• \( y = b \)

In this equation, whatever the x-coordinate might be, the y-coordinate will remain constant as \( b \). If a horizontal line passes through the point \((5, 8)\), the equation would be \( y = 8 \). This represents that for any x-value, the y-coordinate will always be 8.

Coordinate Geometry

Coordinate geometry, sometimes known as analytic geometry, is a branch of mathematics that uses algebraic equations to describe geometric shapes and relationships. It allows us to represent geometric figures like lines, circles, and polygons in a coordinate plane using numerical expressions.

This field is powerful because it combines elements of algebra and geometry thus making it a critical tool for solving many types of problems. **In coordinate geometry,**

• Points are defined by coordinates \((x, y)\).
• Lines can be represented by simple equations that express relationships between x and y.

For instance, the point \((5, 8)\) is a location in the plane, and from this, we can derive specific equations that describe both vertical and horizontal lines passing through it. This intersection of algebra and geometry simplifies the solution of complex problems and provides a clear, visual method of interpretation in math.