Problem 25 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The equation is \(y = 5\).

Step 1: Understanding Parallel Lines

A line that is parallel to the x-axis has a slope of 0 because it runs horizontally. Every point on such a line has the same y-coordinate.

Step 2: Determine Y-Coordinate for the Line

Given that the line passes through the point \((4,5)\), the y-coordinate for every point on the line is 5. This is because parallel lines to the x-axis do not change in y-coordinate.

Step 3: Write the Equation of the Line

Since the line is parallel to the x-axis, the equation takes the form of \(y = b\), where \(b\) is the y-value of any point the line passes through. Therefore, the equation is \(y = 5\).

Key Concepts

Parallel Lines

Parallel lines are fascinating in geometry due to their unique relationship with each other. By definition, parallel lines are two or more lines in the same plane that never intersect, no matter how far they are extended.

In the context of coordinate geometry, a line parallel to either the x-axis or y-axis has particular characteristics:

• **Parallel to the x-axis:** This type of line moves horizontally. It means they keep a constant distance from the x-axis throughout. Their slope is 0, representing a perfectly flat line. The equation of such a line can be expressed as \( y = b \), where \( b \) is the y-coordinate which remains unchanged. For example, in the exercise, the given point is \((4,5)\), which determines \( b = 5 \), hence \( y = 5 \).
• **Parallel to the y-axis:** These lines run vertically, which means they maintain a consistent distance from the y-axis. Such lines do not have a defined slope, often described as undefined or infinite. Their equation is generally expressed as \( x = a \), where \( a \) is the x-coordinate.

Understanding these concepts is key when dealing with parallel lines in coordinate geometry. Recognizing their path and pattern makes it much simpler to work with various problem-solving strategies.

Slope

The slope of a line is a fundamental concept in understanding the behavior of straight lines in coordinate geometry. A line's slope can be thought of as a measure of its steepness or the rate at which it rises or falls.

The slope (\( m \)) of a line connecting two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated as:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

This formula gives us an insight into how much the y-value changes for a given change in the x-value.

In coordinate geometry:

• **Horizontal lines**, like those parallel to the x-axis, have a slope of \( m = 0 \), indicating no rise or fall over any stretch of the line.
• **Vertical lines** have an undefined slope because they run straight up and down, which would involve a division by zero in our slope calculation.
• A line moving upwards from left to right has a positive slope, while one that declines from left to right has a negative slope.

Recognizing these slope types helps in graphing lines quickly and understanding their orientation in the coordinate plane.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that uses coordinates to represent geometric figures. It allows the study of geometry using a formal algebraic framework.

Some key aspects of coordinate geometry include:

• **Plotting Points:** Every point in coordinate space is defined by its position obtained through the x-coordinate and y-coordinate. The point \((x, y)\) tells us how far along and how far up or down the point is from the origin \((0, 0)\).
• **The Cartesian Plane:** This is a two-dimensional plane defined by a horizontal axis (x-axis) and a vertical axis (y-axis). Lines and curves can be expressed with mathematical equations in this plane, allowing for precise calculations and measurements.
• **Equations of Lines:** The equation of a line describes all the points that lie on that line. The simplest form is \( y = mx + c \), where \( m \) represents the slope and \( c \) is the y-intercept.

Coordinate geometry combines algebra and geometry, providing powerful tools to solve geometric problems using algebraic methods. By mastering coordinate geometry, one gains a profound advantage in solving a wide array of mathematical problems, whether simple or complex.