Problem 53 - Algebra A Combined Function

Question

Find an equation of each line described. Write each equation in slope- intercept form when possible. Through (-2,-3) , perpendicular to the \(y\) -axis

Step-by-Step Solution

Answer

The equation of the line is \( y = -3 \).

Step 1: Understand the Slope of a Line Perpendicular to the y-axis

A line that is perpendicular to the y-axis is a horizontal line. Horizontal lines have a slope of 0.

Step 2: Identify the Slope-Intercept Form

The slope-intercept form of a line can be written as \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. For a horizontal line, the slope \( m = 0 \), resulting in the form \( y = b \).

Step 3: Use the Given Point to Find the Equation

The line passes through (-2, -3). Since the slope is 0, the y-value remains constant along the line. Hence, the equation of the line is \( y = -3 \).

Key Concepts

Perpendicular Lines

Understanding perpendicular lines is key in geometry. When two lines are perpendicular, they intersect to form right angles, which are 90° angles.
The focus of most exercises about perpendicular lines is often on their slopes. The slope of a line determines its steepness, and two perpendicular lines have slopes that are negative reciprocals of each other. However, in the case of a line being perpendicular to the y-axis, it is an exception.
This scenario results in a horizontal line. Unlike typical perpendicular lines, a horizontal line to the y-axis doesn't display this reciprocal relationship. Instead, it has a zero slope, because it does not rise or fall at all.
Thus, when dealing with the perpendicularity to axes such as the y-axis, consider the unique characteristics of horizontal and vertical lines, rather than usual perpendicular slope relationships.

Horizontal Line

Horizontal lines are fascinating because they run parallel to the x-axis and have no vertical change as they move along the x-axis.
Such lines are defined by having a slope of zero. The concept of slope involves how steep a line is; with zero slope, horizontal lines are completely flat, indicating no incline or decline.
Usually represented by the equation form:

• \(y = b\)

where \( b \) is the constant y-value for all points on the line. For example, if a line goes through the point (-2, -3), it is simply expressed as \( y = -3 \).
This indicates regardless of the x-values, the y-value remains constant, forming a straight line along the horizontal plane.

Y-Intercept

The y-intercept is a crucial concept when dealing with equations of lines in the slope-intercept form, \( y = mx + b \). In simple terms, the y-intercept is the point where the line intersects the y-axis.
This point occurs when the x-coordinate is zero, allowing the equation to solely reflect the value of \( b \).
For any horizontal line, all points share the same y-coordinate, meaning the entire line is defined by a singular y-intercept.
This concept might seem subtle but is very important in identifying the position of the line on a graph. For example, in the equation \( y = -3 \), the y-intercept is \(-3\), showing that the line crosses the y-axis at this point.
Understanding y-intercepts helps visually represent linear equations and provides insight into how horizontal lines sit on a coordinate plane without ever inclining or declining.