Problem 7 - The Calculus with Analytic Geometry

Question

In Exercises 5 through 14, find an equation of the line satisfying the given conditions. $$ \text { Through the point }(-3,-4) \text { and parallel to the } y \text { axis. } $$

Step-by-Step Solution

Answer

The equation is \( x = -3 \).

Step 1: Understand the problem

We need to find the equation of a line that goes through the point (-3, -4) and is parallel to the y-axis.

Step 2: Identify properties of a line parallel to the y-axis

A line parallel to the y-axis has the same x-coordinate for all points on the line.

Step 3: Write the equation

Since the line passes through (-3, -4) and is parallel to the y-axis, the equation will be in the form \( x = k \), where \( k \) is the x-coordinate of the given point.

Step 4: Substitute the given x-coordinate

Substitute the x-coordinate of the given point (-3, -4) into the equation form \( x = k \). Therefore, the equation is \( x = -3 \).

Key Concepts

Line Parallel to Y-axis

To understand equations of lines parallel to the y-axis, we need to know their unique properties. These lines run vertically across the graph. No matter which point you pick on the line, the x-coordinate will always be the same. This means that if a line is parallel to the y-axis, it does not lean or slope. Instead, it stands straight up and down with a constant x-value.
For example, if a line is parallel to the y-axis and crosses through the x-coordinate of -3, every point on this line will have a corresponding x-value of -3. The y-coordinates along the line can be any number, but the x-coordinate remains constant. Understanding this will make it easier to write the equation for such lines.

Coordinates

Coordinates are essential in defining the position of a point on a graph. They are written in the form of (x, y). The first number, x, represents the horizontal distance from the origin. The second number, y, represents the vertical distance. In our exercise, we are given the point (-3, -4). This means our point is three units to the left of the y-axis and four units down from the x-axis.
In the context of lines, coordinates become particularly important when determining the equation of the line. For a line parallel to the y-axis passing through (-3, -4), we only care about the x-coordinate, as all points on the line will have that same x-value.

Equation Form

Equations provide a mathematical way to express relationships between variables. When dealing with a line parallel to the y-axis, the equation takes a simple form. Instead of the standard form of a line equation (like y = mx + b), lines parallel to the y-axis are written as x = k. Here, 'k' is the constant x-coordinate for all points on the line.
In the given exercise, since the line goes through the point (-3, -4), the x-coordinate we use is -3. Therefore, the equation of the line parallel to the y-axis and passing through (-3, -4) is x = -3.
Remember: when dealing with vertical lines, the equation will always be in the form x = [some constant x-value].