Problem 55 - Algebra A Combined Function

Question

Write an equation of each line. Write the equation in standard form unless indicated otherwise. See Examples 1 through \(6 .\) Through (5,-6)\(;\) perpendicular to \(y=9\)

Step-by-Step Solution

Answer

The equation is \(x = 5\).

Step 1: Understanding Line Perpendicular to a Horizontal Line

The line described in the problem is perpendicular to the line defined by the equation \(y = 9\). Since \(y = 9\) is a horizontal line, any line that is perpendicular to a horizontal line is a vertical line. Vertical lines have undefined slopes and are written in the form \(x = c\), where \(c\) is a constant.

Step 2: Identifying Equation of the Perpendicular Line

Since vertical lines are given in the form \(x = c\), to determine the exact line through the point \((5, -6)\), we simply use the x-coordinate of the given point. Thus, the equation of the line perpendicular to \(y = 9\) and passing through \((5, -6)\) is \(x = 5\).

Key Concepts

Vertical Lines

Vertical lines are unique in the world of geometry. They stand straight up and down, like a skyscraper. Unlike most lines, vertical lines do not have a slope in the way most people think about it.
Instead, their slope is considered "undefined." This is because a slope is typically calculated as the rise over the run, or the change in y divided by the change in x.
However, for vertical lines, the x-value remains constant while the y-values can change dramatically, giving a denominator of zero in the slope formula, creating an undefined situation.
Here are some simple tips to understand vertical lines even better:

• Vertical lines are represented by equations of the form \( x = c \) where \( c \) is a constant.
• These lines run parallel to the y-axis on a graph.
• They do not intersect the x-axis, except at the point where the equation is \( x = c \).
• Vertical lines are used to describe situations where every y-value corresponds to one fixed x-value.

Horizontal Lines

Horizontal lines stretch across the page, never rising or falling. They're great examples of how consistent the y-value can remain in a world of changing x-values.
This consistency gives horizontal lines a slope of zero, which means there is no rise no matter the amount of run.
The slope, often calculated as \( \frac{\text{rise}}{\text{run}} \) or \( \frac{\Delta y}{\Delta x} \), remains zero as the numerator (the rise) is always zero.Consider these essential characteristics about horizontal lines:

• They are expressed in the form \( y = c \) where \( c \) remains constant.
• These lines run parallel to the x-axis.
• They never intersect the y-axis, except at the constant y-value \( y = c \).
• They indicate scenarios where the x-values can vary, but the y-value remains unchanged in linear equations.

Equation of a Line

Understanding the equation of a line is fundamental in coordinate geometry. Regardless of how it's presented, an equation maps out every single point on the line.Lines in the coordinate plane can be expressed in different forms:

• Standard Form: The equation has the structure \( Ax + By = C \), where \( A, B, \) and \( C \) are integers, and \( A \) is non-negative.
• Slope-Intercept Form: This is \( y = mx + b \), where \( m \) is the slope of the line and \( b \) is the y-intercept, or the point where the line crosses the y-axis.
• Point-Slope Form: Used when you know a point on the line \((x_1, y_1)\) and the slope \(m\), given as \( y - y_1 = m(x - x_1) \).

The equation of a line encapsulates how every x and y pair relate to each other.
Knowing these relationships helps in understanding and predicting how a line behaves across a coordinate plane. From testing line parallelisms to determining perpendicular lines like in the example, mastering equations is an essential skill.