Problem 54 - Algebra 1

Question

Write an equation in standard form of the horizontal line and the vertical line that pass through the point. $$(1,3)$$

Step-by-Step Solution

Answer

The equation of the horizontal line passing through the point (1,3) is \(y = 3\). The equation of the vertical line passing through the same point is \(x = 1\).

Step 1: Equation of Horizontal Line

Firstly, let's write the equation for a horizontal line passing through the point (1,3). A horizontal line has all its y-coordinates the same. So, for a specific point (1,3), the y-coordinate will always be 3. Therefore, the equation of the horizontal line is \(y = 3\).

Step 2: Equation of Vertical Line

Moving on, let's find the equation for a vertical line through the point (1,3). In the case of a vertical line, all x-coordinates are the same. So, for our specific point (1,3), the x-coordinate will always be 1. Hence, the equation for the vertical line is \(x = 1\).

Key Concepts

Horizontal Lines

A horizontal line is a straight line that moves left to right across a graph. In simpler terms, it’s a flat line that doesn't go up or down. These lines are unique in coordinate geometry because they have a constant y-coordinate for every point on the line.

• The characteristic equation of a horizontal line is written as: \[y = b\]
• Here, \(b\) represents the y-coordinate where the line crosses the y-axis. No matter the value of \(x\), \(y\) remains \(b\).

Understanding horizontal lines can be useful in analyzing graphs and spotting data trends. In coordinate geometry, a horizontal line might indicate something unchanged or consistent across an axis.

Vertical Lines

In contrast to horizontal lines, vertical lines run from top to bottom on a graph. These lines are characterized by having an unchanging x-coordinate, regardless of the y-coordinate. They are essential in coordinate geometry when addressing equations of lines.

• The representative equation for a vertical line is: \[x = a\]
• Here, \(a\) represents the constant x-value where the line touches the x-axis. Every point along this line has an x-coordinate of \(a\), making them parallel to the y-axis.

Vertical lines can appear in graphs to show constraints or boundaries. Recognizing these lines helps you identify linear relationships acting strictly along one coordinate without variation.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, provides a crucial link between algebra and geometry through graphs and equations. It helps us understand positions and shapes in a two-dimensional space using the coordinate plane.The coordinate plane is defined by two axes:

• x-axis: The horizontal line representing all possible values of \(x\).
• y-axis: The vertical line showing all potential values of \(y\).

Using these, each point on the plane is defined by an ordered pair \((x, y)\), indicating its location. Through coordinate geometry:

• We can derive linear equations, such as horizontal (\(y = b\)) or vertical (\(x = a\)) lines.
• Determine slopes and intercepts which tell us about the direction and position of lines.

This field is indispensable for interpreting graphically represented data. It not only allows visualization of algebraic equations but also serves as a fundamental tool for solving real-world problems.