Problem 24 - Algebra A Combined Function

Question

Write an equation of each line. See Examples 3 and \(4 .\) Horizontal; through (0,5)

Step-by-Step Solution

Answer

The equation is \( y = 5 \).

Step 1: Understanding Horizontal Lines

First, let's understand what a horizontal line means. A horizontal line is a line where all points on the line have the same y-coordinate. This means the line runs parallel to the x-axis.

Step 2: Identify the y-coordinate

We are given the point (0, 5) through which the line passes. For a horizontal line, this point tells us that the y-coordinate for all points on the line will be 5.

Step 3: Write the Equation

Since every point on the line has the same y-coordinate, we write the equation of the line with that constant y value. Hence, the equation of the line is simply the y-coordinate: \[ y = 5 \].

Key Concepts

Horizontal Lines

Horizontal lines are one of the simplest types of lines you might encounter in geometry. These are special because they do not tilt up or down; they stay perfectly flat and run parallel to the x-axis of a graph. This means every point on a horizontal line shares the exact same y-value, regardless of what the x-value is.

For example, if a horizontal line passes through the point (0, 5), then every other point on this line will have the y-coordinate of 5 as well. You could pick any x-value, such as 1, 3.5, or even 100,000, and on this horizontal line, the corresponding y-value will remain 5. So, for a horizontal line, knowing one point on the line tells you the y-value for all points. This makes forming an equation for horizontal lines a straightforward task.

Coordinate Geometry

Coordinate geometry, often referred to as analytic geometry, is the study of geometry using a coordinate system. Through coordinate geometry, we can represent geometric shapes numerically and calculate various properties such as distance, angles, and the slopes of lines.

The coordinate system consists of two perpendicular lines, the x-axis (horizontal) and the y-axis (vertical), that divide the plane into quarters. Each point in this plane can be represented by an ordered pair (x, y). The x-coordinate measures the position to the left or right of the y-axis, while the y-coordinate measures the height above or below the x-axis.

Understanding how to plot points and form lines on a coordinate plane is fundamental. It allows transition from abstract geometric concepts to concrete numerical computations. For example, identifying that a line is horizontal tells us quickly that the line's slope is zero and that any equation for the line will be of the form y = constant. This connection between algebra and geometry offers insightful ways to solve problems.

Writing Equations

Writing equations of lines is a fundamental skill in coordinate geometry. Equations let us describe lines mathematically, so we can use them to analyze and solve geometric problems.

For horizontal lines, the equation is particularly simple. Since all points on a horizontal line have the same y-coordinate, the equation takes the form y = c, where c is the constant y-coordinate shared by all points on the line.

• A horizontal line through (0, 5) has the equation y = 5.
• If it passes through (2, -7), the equation would be y = -7.

To write the equation of a horizontal line, simply identify the y-coordinate of any point the line passes through, and set y equal to that constant. This easy-to-remember rule shows the simplicity and beauty of horizontal lines in mathematics. By practicing writing these equations, students become adept at bridging the gap between geometric intuition and algebraic expression.