Problem 29 - Algebra and Trigonometry

Question

Find an equation of the line that satisfies the given conditions. Through \((-1,2) ;\) parallel to the line \(x=5\)

Step-by-Step Solution

Answer

The equation of the line is \(x = -1\).

Step 1: Identify the Type of Line

The given line is \(x = 5\), which is a vertical line. A vertical line has an undefined slope.

Step 2: Understand Line Parallelism

To find a line parallel to \(x = 5\), note that parallel lines to a vertical line are also vertical. Hence, the new line must also be vertical.

Step 3: Use the Given Point to Define the Line

Since the line must pass through the point \((-1, 2)\), a vertical line through this point has the equation \(x = -1\).

Key Concepts

Parallel Lines

Parallel lines are lines in a plane that never meet; they are always the same distance apart. A key aspect to understand when discussing parallel lines is that if two lines are parallel, they will have the same slope, if they are not vertical lines. In the context of vertical lines, this concept shifts slightly.
When dealing with vertical lines, like the line represented by the equation \(x = 5\), the slope is considered undefined, and parallelism is defined by orientation. This means that any line that is parallel to \(x = 5\) will itself be a vertical line. For example, a line parallel to \(x = 5\) which passes through \((-1, 2)\) would be \(x = -1\).
When you encounter a problem involving line parallelism, always check the orientation of the given line. If it's horizontal or has a defined slope, use the same slope for the parallel line. If it's vertical, find the x-value the new line will pass through to maintain that parallel condition.

Vertical Lines

A vertical line is a line where all points on it have the same x-coordinate. Its general form is \(x = c\), where \(c\) is a constant. These lines run up and down the graph rather than left to right. An important thing to note is that vertical lines are not functions because they do not pass the vertical line test—a test that checks whether a line touches a graph more than once at any given x-coordinate.
Vertical lines have several distinct properties:

• They do not intercept the x-axis.
• They have an undefined slope.
• They are denoted simply by \(x =\) followed by a fixed number.

For example, in the original exercise, the line \(x = 5\) represents a vertical line. A parallel vertical line that passes through the point \((-1, 2)\) would naturally be \(x = -1\), maintaining the structure and properties of vertical lines.

Undefined Slope

Slope is a measure of how steep a line is, usually described as a rise over run. However, for vertical lines, this concept becomes tricky. In mathematical terms, a vertical line means you're dividing by zero to calculate slope, which is impossible. Hence, vertical lines are said to have an undefined slope.
When graphing a line or trying to write its equation, the slope is crucial if the line isn't vertical. For vertical lines, the notion of an undefined slope highlights that these lines rise (or fall) without running left or right.
Understanding the concept of an undefined slope is essential when working with equations that define vertical lines. This term simply signifies that the usual method for calculating slope doesn't apply, but knowing this helps to anticipate the behavior and properties of these lines when solving geometric problems.