Problem 30 - Algebra and Trigonometry

Question

Find an equation of the line that satisfies the given conditions. Through \((4,5) ;\) parallel to the \(y\) axis

Step-by-Step Solution

Answer

The equation is \(x = 4\).

Step 1: Identify the direction

Since the line is parallel to the y-axis, it is vertical. A vertical line has the form \(x = k\), where \(k\) is a constant.

Step 2: Determine the value of k

Given that the line passes through the point \((4,5)\), the x-coordinate of any point on the line will be 4. Therefore, \(k = 4\).

Step 3: Write the equation

Now that we know the form of the equation and the value of \(k\), the equation of the line is \(x = 4\).

Key Concepts

Understanding Vertical Lines

Vertical lines are a fundamental concept in geometry and algebra. Unlike diagonal or horizontal lines, a vertical line moves straight up and down. This means it doesn't have a slope like other lines. Instead, vertical lines are defined by their position along the x-axis. Every point on the same vertical line shares the same x-coordinate. This is why the equation of a vertical line is simply in the form of \(x = c\), where \(c\) is a constant representing the x-coordinate for all points on the line.
This distinct characteristic makes vertical lines unique and easily recognizable in the context of coordinate geometry. Remember, no matter what the y-coordinate is, for a vertical line, all x-coordinates will be identical.

Lines Parallel to the Y-Axis

When a line is parallel to the y-axis, it precisely means that the line is vertical. Being parallel to the y-axis implies that the line doesn't intersect the y-axis at any point, except if it's to the left or right of it, maintaining a consistent distance.

• A vertical line that runs parallel to the y-axis will have an x-coordinate that remains constant for all its points.
• This is because the line moves up and down but never left or right, preserving its alignment parallel to the y-axis.

In practical terms, if you're looking to determine a line that is parallel to the y-axis, you're essentially identifying a vertical line with a constant x-value, precisely like the line in our exercise where the equation is \(x = 4\).

Using Point-Slope Form with Vertical Lines

The point-slope form is a helpful equation for finding the equation of a line. Generally, it's expressed as \(y - y_1 = m(x - x_1)\). However, vertical lines are an exception here, because the concept of slope doesn't apply in the traditional sense.
Here's why vertical lines can't use the point-slope form:

• Slope (\(m\)) is calculated as "rise over run," or change in y over change in x. For vertical lines, the change in x is zero, making the slope undefined.
• Therefore, you can't use the regular point-slope form without running into mathematical errors.

Instead, for vertical lines, simply focus on using the x-coordinate from the point to write the equation: \(x = x_1\), where \(x_1\) is the x-coordinate of the point the line goes through, just like in the given exercise with \(x=4\).