Problem 48 - Algebra A Combined Function

Question

Write an equation of each line. Write the equation in standard form unless indicated otherwise. See Examples 1 through \(6 .\) Horizontal line; through (1,0)

Step-by-Step Solution

Answer

The equation of the horizontal line is \(y = 0\) or in standard form \(0x + 1y = 0\).

Step 1: Understanding the Problem

The exercise asks us to find the equation of a horizontal line that passes through the point \((1,0)\). Horizontal lines have the same y-coordinate for all x-values.

Step 2: Identifying Line Properties

A horizontal line has the form \(y = b\), where \(b\) is the y-coordinate of any point the line passes through. In this case, the point is \((1, 0)\), so \(b = 0\).

Step 3: Writing the Equation

Since we know horizontal lines are expressed as \(y = b\), and from the point \((1, 0)\) we have \(b = 0\), the equation of the line is \(y = 0\).

Step 4: Converting to Standard Form

The standard form for a line's equation is \(Ax + By = C\). The equation \(y = 0\) can be rewritten in standard form by expressing it as \(0x + 1y = 0\). Therefore, the standard form of the line is \(0x + 1y = 0\).

Key Concepts

Horizontal Line

A horizontal line is a type of linear equation that displays a unique characteristic: it has a constant y-coordinate across all its points. This means that no matter the x-value, the y-value remains unchanged. In the Cartesian coordinate plane, horizontal lines run parallel to the x-axis.
If you imagine drawing a straight line across the plane, it would never go up or down, just left or right. An easy way to recognize a horizontal line is by its equation form, which is always written as \( y = b \). Here, \( b \) represents the y-coordinate of any point on that line.

In simpler terms, if given a point, such as \((1, 0)\), the equation of the line passing through that point will always be \( y = 0 \), since the y-coordinate at all points along the line is zero. No calculations for the x-values are needed! This makes horizontal lines quite straightforward to deal with in algebra.

Standard Form

The "standard form" of a linear equation is a specific way of expressing any line on a coordinate plane. It is written in the format \( Ax + By = C \), where \( A \), \( B \), and \( C \) are integers, and \( A \) and \( B \) are not both zero.
Standard form is a versatile way to express any line, including horizontal or vertical lines, and is often used in algebra to make working with equations easier.

When converting a simple horizontal line equation, such as \( y = 0 \), to standard form, the process is simple. Since there is no x-term, it becomes \( 0x + 1y = 0 \). This conversion takes the concept of a horizontal line and integrates it into a format that can be more easily manipulated and interpreted in various algebraic operations.

Coordinates

Coordinates are fundamental in geometry, especially when dealing with equations of lines. They are an ordered pair, typically written in the form \((x, y)\), that locate a point's position on a coordinate plane. The first value \( x \) tells you how far to move horizontally, whereas the second value \( y \) tells you how far to move vertically.
These coordinates are crucial to defining lines, as any line can be determined by at least one point it passes through.

Consider the point \((1, 0)\). This tells you that the point is 1 unit away from the y-axis and right on the x-axis. For horizontal lines in particular, the significant part of the coordinate is the y-value, as it tells you the line's height (or lack thereof) across the entire x-axis. Thus, understanding coordinates can deeply enrich your understanding of how lines behave and are represented in algebra and geometry.