Problem 50 - College Algebra with Modeling and Visualization

Question

Find an equation of the line satisfying the conditions. Vertical, passing through \((1.95,10.7)\)

Step-by-Step Solution

Answer

The equation of the line is \( x = 1.95 \).

Step 1: Understanding Vertical Lines

A vertical line in the coordinate plane is a line that moves up and down and does not have a slope. The equation of a vertical line is of the form \( x = a \), where \( a \) is a constant that represents the x-coordinate of any point on the line.

Step 2: Identify x-coordinate of the Point

Since the line is vertical and we know it passes through the point \((1.95, 10.7)\), our x-coordinate here is \(1.95\). This means every point on this vertical line will have an \( x \)-value of \(1.95\).

Step 3: Write the Equation of the Vertical Line

Given the x-coordinate of the point is \(1.95\), the equation of the line will be \( x = 1.95 \), as the line is vertical.

Key Concepts

Coordinate Plane

Every time you explore mathematics involving graphs and lines, you're often dealing with the coordinate plane. Think of it as a giant grid where you can easily pinpoint any location using two numbers: the x-coordinate and the y-coordinate. These two coordinates are what mathematicians call an "ordered pair," usually written as \((x, y)\).

• The coordinate plane is divided into four sections called quadrants, which help us determine the signs of the x and y values.
• The horizontal axis is the x-axis, running left and right, while the vertical axis is the y-axis, running up and down.

Visualizing the coordinate plane allows us to understand positions and movements, like the path of a line or curve, much more clearly and is foundational in graphing equations effectively. Each point is plotted specifically according to its x and y values, giving a clear depiction of its location.

Equation of a Line

In a world filled with straight lines, knowing how to express them mathematically is key. The equation of a line captures all the points that lie on a given line. Depending on the orientation of the line (vertical, horizontal, or slanted), the equation can take different forms.

• For most lines that aren't vertical, the equation commonly takes a slope-intercept form, written as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
• For a vertical line, no slope exists because it does not tilt to the left or right. Thus, its equation is simply \(x = a\).

So when you encounter a vertical line, you're dealing with an equation reflecting all points having the same x-coordinate. This vertical nature is special, and it stands out from slanted lines due to its simplicity in equation form.

X-coordinate

The x-coordinate is a fundamental part of any point's identity in the coordinate plane. It tells you exactly how far along the horizontal axis a point is located. In the point \((x, y)\), the x-coordinate is the first number. It's an integral part of determining the position of any point.

• For vertical lines, every point on the line shares the same x-coordinate, making these lines unique because their equations simply state \(x = \) some constant.
• The value of the x-coordinate can also help identify the specific horizontal section of the plane in which the point lies.

Understanding x-coordinates helps solve problems like finding whether a given point belongs to a line or determining the equation of a vertical line. Through the x-coordinate, you can quickly express and recognize vertical lines on the plane.