Problem 49 - Elementary Algebra

Question

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: Understanding the Line Parallel to the y-axis

A line that is parallel to the y-axis is a vertical line. The equation of a vertical line is of the form \( x = a \), where \( a \) is a constant value representing the x-coordinate of any point on the line.

Step 2: Identify the x-coordinate for the line

Since the line must pass through the point \((2,-4)\), we set the x-coordinate equal to the x-value of this point. Thus, the equation of the line is \( x = 2 \).

Step 3: Express in Standard Form

Standard form typically refers to a linear equation in the form \( Ax + By = C \). However, for a vertical line, the equation \( x = 2 \) is already in its simplest form and adequately represents the standard form for a line parallel to the y-axis as \( x = C \).

Key Concepts

Vertical Lines

Vertical lines are unique in algebra due to their distinct characteristics. Unlike other lines, which may slant upward or downward on a graph, vertical lines maintain a consistent direction. They go straight up and down, parallel to the y-axis. Because of this property, the equation of a vertical line is always in the form \( x = a \). Here, \( a \) is a constant number, representing the x-coordinate for any and all points that lie on the line.

This means for any vertical line, its equation is derived directly from the x-coordinate of a point the line passes through. This simplifies the understanding and plotting of vertical lines, as you only need one value - the x-coordinate. In the context of our example, the vertical line passes through the point \((2, -4)\) thus, its equation is \( x = 2 \). Vertical lines are exceptions when discussing slope since their slope is undefined.

Point-Slope Form

Understanding the point-slope form of a line is a crucial part of algebra. This form is particularly useful for writing linear equations when you know a point on the line and its slope. The point-slope form is represented as: \( y - y_1 = m(x - x_1) \). Here, \((x_1, y_1)\) is a known point on the line and \( m \) is the slope.

Whenever you have a non-vertical line and know any point it passes through along with the slope, you can substitute these values into the equation to find the line's equation. However, when a line is vertical, like in our problem, the slope is undefined, making the point-slope form not applicable. Instead, vertical lines are best expressed in their own form, \( x = a \), rendering point-slope form unnecessary. Though not usable here, understanding point-slope form could be instrumental in situations involving non-vertical lines.

Standard Form in Algebra

In algebra, the standard form of a linear equation is expressed as \( Ax + By = C \). "A", "B", and "C" are constants, where "A" should be non-negative and both "A" and "B" are not zero. The standard form is beneficial when needing a general representation of a line, especially for computational ease or when comparing slopes and intercepts.

Vertical lines, however, present a special case in standard form. In typical circumstances, a vertical line is simply written as \( x = a \), lacking the "y" component entirely because the slope is undefined. This simple form can be seen as a variation of the standard form where "B" equals zero: \( 1x + 0y = a \).

For example, in our exercise, since the line is vertical and goes through \( (2, -4) \), its equation is \( x = 2 \). Despite its simplicity, this form effectively communicates that the line runs parallel to the y-axis, making the equation straightforward and easy to understand.