Problem 29 - Precalculus Mathematics for Calculus

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: Understand the Conditions

The line must pass through the point \((4, 5)\) and be parallel to the \(x\)-axis.

Step 2: Characteristics of a Line Parallel to the X-Axis

A line parallel to the \(x\)-axis has a constant \(y\)-coordinate for all points on the line. Its equation has the form \(y = c\), where \(c\) is a constant.

Step 3: Identify the Constant Y-Value

Since the line must pass through \((4, 5)\), the \(y\)-coordinate is 5. This means \(c = 5\) in the equation form \(y = c\).

Step 4: Write the Equation

Substitute \(c = 5\) into the equation \(y = c\). Thus, the equation of the line is \(y = 5\).

Key Concepts

Parallel Lines

Parallel lines are a key concept in coordinate geometry. They are lines that run alongside each other at a constant distance, never meeting or intersecting, no matter how far they are extended. Parallel lines are identified by their slopes; they have the exact same slope but different y-intercepts.
For instance, consider two parallel lines in the coordinate plane described by the equations \(y = 2x + 3\) and \(y = 2x - 4\). Both lines have a slope of 2, making them parallel, but their y-intercepts differ (3 and -4, respectively).
This concept applies to our exercise since a line parallel to the x-axis is special in that its slope is 0. This means it doesn't tilt up or down, just straight across horizontally. So, when reviewing any problem, looking for matching slopes is a great way to determine if lines are parallel.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. It allows for the description of geometric shapes numerically and the extraction of algebraic equations from geometric figures. In this space, every point can be identified by an ordered pair of numbers called coordinates, typically represented as \((x, y)\).
These coordinates correspond to the horizontal (x-axis) and vertical (y-axis) placement of a point on a plane. Understanding this is crucial for plotting lines or shapes, solving equations, or verifying geometric properties. In our exercise, the point \((4, 5)\) indicates the position of the line parallel to the x-axis. The x-coordinate (4) tells us how far along from the vertical y-axis the point is, and the y-coordinate (5) tells us how high up from the horizontal x-axis it is.
Using this system, we can describe the line as a collection of points that all share the same y-value and differ only in their x-values, forming a straight line parallel to the x-axis.

Linear Equations

Linear equations are equations of the first order and involve no terms raised to powers greater than one. They graph as straight lines, hence the term 'linear'. These equations take various forms, but the simplest is \(y = mx + b\), where \(m\) stands for the slope and \(b\) indicates the y-intercept:

• The slope \(m\) describes how steep the line is.
• The y-intercept \(b\) tells us where the line crosses the y-axis.

Linear equations are foundational in solving various problems in mathematics and science, as they provide a straightforward relation between variables.
In our particular exercise, we're concerned with a special kind of linear equation: one that's entirely horizontal. This horizontal line doesn't slant up or down, indicating that its slope is 0. Consequently, the equation simplifies from \(y = mx + b\) to \(y = c\), where \(c\) is a constant. This reflects a line's characteristic of having the same y-value across all plotted points, which speaks directly to our line being parallel to the x-axis with an equation \(y = 5\).