Problem 37 - Intermediate Algebra

Question

Contains the point \((2,-4)\) and is parallel to the \(y\) axis

Step-by-Step Solution

Answer

The equation is \(x = 2\).

Step 1: Understanding the Problem

We need to find the equation of a line that contains the point \((2, -4)\) and is parallel to the \(y\)-axis.

Step 2: Identifying Line Characteristics

A line parallel to the \(y\)-axis has an undefined slope and is a vertical line. The equation of a vertical line is of the form \(x = a\), where \(a\) is a constant value.

Step 3: Applying the Point to the Equation

Since the line must pass through the point \((2, -4)\), the constant \(a\) in the equation \(x = a\) is equal to the \(x\)-coordinate of the point. Thus, the equation is \(x = 2\).

Step 4: Verifying the Line Equation

Verify that the line \(x = 2\) passes through the point \((2, -4)\). In this case, since the equation implies that \(x=2\) for all points, and the point has \(x=2\), it satisfies the equation.

Key Concepts

Parallel Lines

Parallel lines are lines in a plane that never meet. They are always the same distance apart and have the same slope when graphed in a Cartesian plane. When we talk about lines parallel to the y-axis, however, they are vertical lines. Vertical lines have an undefined slope because their rise is infinite, making it impossible to calculate a numerical slope. This unique nature distinguishes vertical lines from other lines that can be expressed in the slope-intercept form, which is why vertical lines are described with equations of the form \(x = a\). In this format, the equation simply states that \(x\) is always a constant value, guaranteeing the line does not deviate in the horizontal direction, thus staying parallel to the y-axis. The consistency of this form is what makes vertical lines readily identifiable, regardless of where they appear on the graph.

Equation of a Line

The equation of a line is a mathematical expression that describes all the points along the line in a plane. The most common form is the slope-intercept form, written as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. However, vertical lines do not fit the slope-intercept model because they have an undefined slope. Instead, the equation of a vertical line is \(x = a\), where \(a\) is the x-coordinate where the line intersects the x-axis. This reflects the nature of a vertical line, indicating it runs up and down along a constant x-value. Vertical line equations are simple and straight to the point. They provide a valuable example of how different kinds of lines require specific equations, showcasing the diversity of line representation in mathematics.

Coordinates and Points

Coordinates are critical in defining the precise location of points in a plane. In mathematics, coordinates are written as pairs, like \((x, y)\), where \(x\) is the horizontal position and \(y\) is the vertical position. In the context of a vertical line, only the x-coordinate has significance, as the line's equation is \(x = a\). For instance, a line passing through \((2, -4)\) and parallel to the y-axis has the equation \(x = 2\). Here, the x-coordinate tells us exactly where the line runs vertically. Understanding coordinate pairs helps us interpret what a line or point represents in space. It also allows us to solve problems involving line equations effectively by applying specific x or y values to verify points on a line. This concept also illustrates the dual role coordinates play in both identifying and checking points on a line.