What's The Equation Of A Line That Passes Through Points (1, 5) And (1, 3)? Question 20 Options: Y=-2xy=1/4x-2y=-x+4y=4x-1

What's The Equation Of A Line That Passes Through Points (1, 5) And (1, 3)? Question 20 Options: Y=-2xy=1/4x-2y=-x+4y=4x-1

When working with coordinate geometry, one of the fundamental questions students and mathematicians often encounter is: "What is the equation of a line that passes through two given points?" In this case, the points are (1, 5) and (1, 3). These points are particularly interesting because they share the same x-coordinate, which indicates that the line passing through them is vertical. Understanding how to determine the equation of such a line, especially when multiple options are provided, is crucial for mastering algebra and coordinate geometry concepts.

In this article, we will explore the process of finding the equation of a line passing through points (1, 5) and (1, 3), analyze the given options, and provide a comprehensive understanding of the underlying principles involved. Whether you're a student studying for an exam or someone interested in the fundamentals of geometry, this guide aims to clarify these concepts in detail.

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Understanding the Basics: Coordinates and Lines

Before diving into the problem, let's revisit some fundamental concepts:

Coordinates of a Point

• A point in a coordinate plane is represented as (x, y).
• The x-coordinate indicates horizontal position.
• The y-coordinate indicates vertical position.

Line in Coordinate Geometry

• A line is a straight one-dimensional figure extending infinitely in both directions.
• Its equation relates x and y coordinates of all points lying on it.

Types of Lines Based on Coordinates

• Horizontal line: All points have the same y-coordinate (e.g., y = c).
• Vertical line: All points have the same x-coordinate (e.g., x = c).
• Oblique (sloped) line: Has a non-zero slope and can be written in various forms (e.g., y = mx + b).

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Analyzing the Given Points: (1, 5) and (1, 3)

Let's look closely at the two points:

• Point A: (1, 5)


• Point B: (1, 3)

Notice that both points have the same x-coordinate, which is 1. This is a critical observation because it indicates that the line passing through these points is vertical.

Implication of Same x-Coordinate

• Since both points share x = 1, the line must be vertical.
• The equation of a vertical line passing through x = 1 is simply x = 1.

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Finding the Equation of the Line

Given the analysis above, the line passing through (1, 5) and (1, 3) is a vertical line at x = 1.

Standard Equation for a Vertical Line

• The general form: x = c, where c is the x-coordinate of every point on the line.
• For our points, since both have x = 1, the equation is x = 1.

Comparison with the Provided Options

The options given are:

• Y = -2
• xy = 1
• 1/4 x - 2
• y = -x + 4
• y = 4x - 1

Let's analyze each:

1. Y = -2: This is a horizontal line at y = -2, which is not relevant here.
2. xy = 1: This is a nonlinear equation, representing a hyperbola, not a line.
3. 1/4 x - 2: This is a linear equation, but with slope 1/4 and y-intercept at -2.
4. y = -x + 4: A line with slope -1, crossing y-axis at 4.
5. y = 4x - 1: A line with slope 4 and y-intercept at -1.

None of these options directly match the vertical line x=1, but the key is to recognize that the actual equation is x=1.

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Why is "x = 1" Not Listed Among the Options?

Given the options, it appears that none explicitly present the vertical line equation x=1. This could be an oversight or a trick in the question. Sometimes, multiple-choice questions may include distractors or options that are not directly related to the actual line.

However, understanding the nature of the line is essential:

• Since both points share the same x-coordinate, the line is vertical at x=1.


• Vertical lines cannot be expressed in slope-intercept form (y=mx + b), because their slope is undefined.

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Understanding the Slope and Equation of a Line

Let's briefly review how to find the equation when the points do not share the same x-coordinate, and then compare that process to our current problem.

General Method for Finding the Equation of a Line Passing Through Two Points

1. Calculate the slope \( m \):

\[ m = \frac{y2 - y1}{x2 - x1} \]

1. Use the point-slope form of the line:

\[ y - y1 = m(x - x1) \]

1. Simplify to slope-intercept form:

\[ y = mx + b \]

In our specific case, because \( x2 - x1 = 0 \), the slope is undefined, indicating a vertical line.

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Special Case: Vertical Lines

Vertical lines are unique because:

• They do not have a slope in the traditional sense (it's undefined).


• Their equation is always of the form x = c, where c is the constant x-value.

In our problem:

\[
x = 1
\]

This is the most precise representation of the line passing through (1, 5) and (1, 3).

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Conclusion: Correct Equation of the Line

Based on the above analysis:

• The line passing through (1, 5) and (1, 3) is vertical.


• Its equation is x = 1.


• None of the options provided explicitly indicate x = 1.

Therefore, if the question asks for the equation of the line passing through these points, the answer should be x = 1.

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Additional Insights and Related Problems

To deepen your understanding, here are some related concepts and problems:

1. Line with Different x-Coordinates

• If the points were (2, 3) and (4, 7), the slope would be:

\[ m = \frac{7 - 3}{4 - 2} = \frac{4}{2} = 2 \]

• Equation using point-slope form:

\[ y - 3 = 2(x - 2) \]

• Simplifies to:

\[ y = 2x - 1 \]

2. Horizontal Line Example

• Points: (5, 2) and (10, 2)
• Equation: y = 2

3. When to Recognize Vertical Lines

• When the x-coordinates are the same.
• Slope is undefined.
• Equation is x = c.

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Summary and Final Remarks

• The points (1, 5) and (1, 3) lie on a vertical line.
• The equation of this line is x = 1.
• This form cannot be expressed as y = mx + b because the slope is undefined.
• Understanding the nature of the points helps in quickly identifying the correct line equation.

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Additional Tips for Solving Similar Problems

• Always check the coordinates for shared x or y values.
• Recognize the difference between vertical and horizontal lines.
• Remember that the slope of a vertical line is undefined.
• When options include equations in slope-intercept form, verify whether the line is horizontal, vertical, or oblique.

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In conclusion, the equation of the line passing through points (1, 5) and (1, 3) is x = 1, representing a vertical line. Recognizing the characteristics of the points is key to determining the correct equation, especially when multiple options are provided. Mastering these concepts will enhance your problem-solving skills in coordinate geometry and prepare you for more advanced mathematical challenges.