Problem 49 - Elementary Algebra

Question

For Problems \(45-60\), write the equation of the line that satisfies the given conditions. Express final equations in standard form. Contains the point \((2,-4)\) and is parallel to the \(y\) axis

Step-by-Step Solution

Answer

The equation of the line is \(x = 2\).

Step 1: Identify Key Characteristics

Since the line is parallel to the y-axis, it is a vertical line. Vertical lines have an undefined slope and run along the x-axis.

Step 2: Use the Point to Determine the Line

The line passes through the point \((2, -4)\). For a vertical line, the x-coordinate is constant. Hence, the equation of the line is \(x = 2\).

Step 3: Express in Standard Form

Standard form of a line is written as \(Ax + By = C\). For the vertical line equation \(x = 2\), it can be expressed as \(1x + 0y = 2\). The coefficients are \(A = 1\), \(B = 0\), and \(C = 2\).

Key Concepts

Standard Form

The standard form in linear equations is a specific way to present a linear equation. This form is helpful because it clearly distinguishes the coefficients and constant. The standard form of a linear equation is written as:

\[ Ax + By = C \]

Where:

• \( A \), \( B \), and \( C \) are integers.
• \( A \) should ideally be a non-negative integer.

The beauty of expressing equations in this form is that it makes certain properties of the line more visible.

For example, the intercepts with the axes can be easily derived. When dealing with vertical lines, the standard form becomes particularly interesting. If a vertical line passes through a point like \( (2, -4) \), the equation is \( x = 2 \), which can be put into standard form as \( 1x + 0y = 2 \). This clearly identifies that the line is vertical with a constant \( x \)-coordinate.

Parallel Lines

Two lines are parallel if they never intersect each other, essentially running alongside each other at a consistent distance. In terms of linear equations, two lines are parallel if they have the same slope.

When a line is parallel to another, the slopes are equivalent. This fact leads us to an interesting concept in geometry—vertical lines, which are a special case.

If a line is described to be parallel to the y-axis, it implies that the line is also vertical. Vertical lines, such as \( x = 2 \), share the property of being parallel to others like \( x = 3 \) due to their undefined slope. These lines run perpendicular to all horizontal lines and extend indefinitely up and down, never crossing the y-axis.

Vertical Lines

Vertical lines are unique in the realm of linear equations. Unlike most lines, vertical lines do not have a well-defined slope. Instead, their slope is considered to be undefined. This is because the rise over run, which determines slope, involves a division by zero since there is no horizontal change.

These lines always take the form of \( x = a \), where \( a \) is a constant. For instance, a line passing through \( (2, -4) \) that runs parallel to the y-axis can be expressed as \( x = 2 \). In this case, for any value \( y \), \( x \) remains constant at 2.

Vertical lines are critical in understanding geometric relationships and coordinate graphing. They do not cross the y-axis and remain parallel to it. In standard form, a vertical line can be expressed as \( Ax + By = C \) with \( B = 0 \), highlighting that there is no dependence on \( y \).