Problem 34 - Algebra A Combined Function

Question

Write an equation of each line. Write the equation in the form \(x=a, y=b\), or \(y=m x+b\). See Examples 5 and \(6 .\) Through (4,-6)\(;\) parallel to \(x=-2\)

Step-by-Step Solution

Answer

The equation of the line is \(x = 4\).

Step 1: Identify Line Characteristics

The line through the point (4, -6) is parallel to the line given by the equation \(x = -2\). This is a vertical line, which means that all parallel lines are also vertical, with undefined slope.

Step 2: Determine Line Form

Since the line is vertical, it must be in the form \(x = a\). Parallel vertical lines maintain the same x-coordinate for any y value, but through the specific x-coordinate of the given point.

Step 3: Use Point to Write Equation

We have the point (4, -6), which tells us the x-coordinate of this vertical line. Thus, the equation of the line is \(x = 4\).

Key Concepts

Parallel Lines

Parallel lines are fascinating in geometry because they never intersect. Think of railroad tracks running alongside each other, always at the same distance apart. This feature means they have the same slope if they are not vertical. However, in the case of vertical lines, the concept of slope changes. Vertical lines have undefined slopes, but they are still parallel to each other.
In this exercise, we see a line described by the equation \(x = -2\), which is vertical. A line through any point parallel to it, such as through (4, -6), will have the same characteristic of keeping the x-coordinate constant. That's why a parallel line through (4, -6) is expressed as \(x = 4\).
Remember:

• Parallel lines have the same gradient or slope.
• Vertical lines have an undefined slope, making them parallel to each other.
• The equation form of a vertical line is \(x = a\).

Vertical Lines

Vertical lines stand tall, running up and down, and they have a unique position in coordinate geometry. Their equation comes in the form of \(x = a\), where \(a\) is the constant x-coordinate for the line. They don't have a slope as such, because the change in x between points is zero, resulting in an undefined slope.
Let's consider why the given problem specifies a line parallel to \(x = -2\). This line, and any line parallel to it, don’t change across the x-axis. They remain vertical. The point (4, -6) helps us identify the specific vertical line passing through it, which results in \(x = 4\).
To summarize:

• Vertical lines are characterized by a constant x value.
• They are all parallel to each other because they share this property of having an undefined slope.
• When given a point like (4, -6), the specific vertical line through it uses that x-coordinate: \(x = 4\).

Coordinate Geometry

Coordinate geometry combines algebra and geometry using a coordinate system. It allows us to represent geometric figures in an algebraic form, like lines, curves, and shapes. A powerful tool in geometry, it uses ordered pairs (x, y) to pinpoint locations and define figures:

• The coordinate plane consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical).
• Points are defined by coordinates (x, y) which convey their exact position on the plane.
• Equations in coordinate geometry provide precise descriptions of lines and curves.

For example, the vertical line \(x = 4\) through point (4, -6) is defined by all points (4, y), where y can be any value. Coordinate geometry uses this framework to express properties and relations, making it easier to visualize and analyze geometric concepts.

Line Equations

Line equations serve as the backbone of coordinate geometry, allowing us to mathematically express and work with lines. They can usually be expressed in different forms, depending on the line's orientation:

• The slope-intercept form: \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
• The vertical line form: \(x = a\), where \(a\) is a constant, representing a line parallel to the y-axis.

In this specific problem, since we are discussing a vertical line, we use \(x = a\). It's simpler than the slope-intercept form because the concept of slope doesn't apply to vertical lines.
To write a line equation efficiently, identify whether the line is vertical or sloped. For vertical lines, use the x-coordinate directly. This exercise shows how easy it can be: given point (4, -6), the line equation is simply \(x = 4\).
Line equations are critical in helping us visualize and solve geometric puzzles algebraically.