Problem 29 - 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 of the line is \(y = 5\).

Step 1: Understanding the Condition

The equation of a line parallel to the x-axis has a constant y-coordinate across all its points. This means that for any line through a point \((x_1, y_1)\) and parallel to the x-axis, the line equation is in the form: \(y = c\) where \(c\) is the y-coordinate of the given point.

Step 2: Extract Y-Coordinate

We need the y-coordinate of the given point, which is \( (4, 5) \). The y-coordinate here is \(5\).

Step 3: Write the Equation of the Line

Since the line is parallel to the x-axis and passes through the point \((4, 5)\), its equation is \(y = 5\). This equation represents a horizontal line passing through all points where the y-coordinate is 5.

Key Concepts

Parallel Lines

In simple terms, parallel lines are lines in a plane that never meet. No matter how far you extend them in either direction, they will never intersect. This is because they have the same slope.For example:

• If two lines have equations, say \(y = 2x + 3\) and \(y = 2x - 5\), they are parallel because they both have the same slope, \(m = 2\).

In terms of relationships:

• The steeper the slope, the more vertical the line is.
• Lines that run left to right and never cross are horizontal or have 0 slopes, making them parallel to the x-axis.

Understanding parallel lines is a fundamental concept for grasping how to determine if two lines are parallel, regardless of whether they are vertical, horizontal, or slanted.

X-axis Parallelism

When we talk about a line being parallel to the x-axis, we're talking about a horizontal line. Such a line maintains the same y-coordinate at every point, just like a shelf that stays perfectly level. The concept is simple:

• A line is parallel to the x-axis if it never rises or falls – it stays level across the graph.
• The slope of a line parallel to the x-axis is 0, as there is no vertical change as you move along the line.

Visualizing this: Imagine stretching a long strip of tape across a wall. If the tape is level – no ups and downs – it's parallel to the floor (x-axis). This means every point on that tape is on the same horizontal plane.

Horizontal Line Equation

A horizontal line has a unique equation form. This is because all points on the line share the same y-coordinate.The rule:

• The equation for a horizontal line is \(y = c\), where \(c\) is a constant.

For instance, a line through the point \(4, 5\) that is horizontal will simply have the equation \(y = 5\).Let's analyze:

• This means regardless of what the x-coordinate is, the y-value on this line will always be 5.
• It creates a flat path on the graph, straight across parallel to the x-axis.

This makes horizontal lines easy to identify and work with in mathematical calculations or real-world applications involving consistent levels.