Parallel Lines In Xy-Plane How To Find Equations Of Parallel Lines

When tackling problems involving parallel lines in the $xy$-plane, a solid grasp of linear equations and their properties is essential. This article aims to provide a comprehensive explanation of how to identify parallel lines based on their equations, focusing on the critical concept of slope. We will delve into the characteristics of parallel lines, explore the slope-intercept form of a linear equation, and dissect the given problem step by step. By the end of this discussion, you'll be equipped with the knowledge and skills to confidently solve similar problems and understand the underlying principles of linear equations.

Core Concepts: Parallel Lines and Slope

The cornerstone of this problem lies in understanding the relationship between parallel lines and their slopes. Parallel lines, by definition, are lines that never intersect. They run in the same direction and maintain a constant distance from each other. This geometric property translates directly into an algebraic characteristic: parallel lines have the same slope. The slope of a line, often denoted by m, quantifies its steepness and direction. It represents the change in the vertical direction (the y-axis) for every unit change in the horizontal direction (the x-axis). A positive slope indicates an upward slant, a negative slope indicates a downward slant, a zero slope represents a horizontal line, and an undefined slope corresponds to a vertical line. The steeper the line, the greater the absolute value of its slope.

To truly grasp this concept, imagine two lines drawn on a graph. If they have the same slope, they will rise or fall at the same rate, ensuring they never converge or diverge. Conversely, if their slopes differ, the lines will eventually intersect. This fundamental principle is crucial for identifying parallel lines from their equations. When presented with multiple linear equations and asked to identify those representing parallel lines, the primary focus should be on comparing their slopes. The y-intercept, which indicates where the line crosses the y-axis, does not play a role in determining parallelism. Lines with the same slope but different y-intercepts will be parallel, while lines with the same y-intercept but different slopes will intersect at that point. This distinction is vital for accurately interpreting linear equations and their graphical representations.

The Slope-Intercept Form: Unveiling the Slope

The slope-intercept form is a standard way to express a linear equation, and it is particularly useful for identifying the slope and y-intercept of a line. This form is written as:

$$y = mx + b$$

where:

• y represents the dependent variable (the vertical coordinate)
• x represents the independent variable (the horizontal coordinate)
• m represents the slope of the line
• b represents the y-intercept (the point where the line crosses the y-axis)

The beauty of the slope-intercept form lies in its clarity. By simply looking at the equation, you can immediately identify the slope as the coefficient of the x term and the y-intercept as the constant term. This direct correspondence makes it incredibly easy to compare the slopes of different lines and determine if they are parallel. For instance, consider the equation y = 3x + 2. In this case, the slope m is 3, and the y-intercept b is 2. This means the line rises 3 units for every 1 unit it moves to the right and crosses the y-axis at the point (0, 2). Now, let's consider another equation: y = 3x - 1. This line also has a slope of 3, but its y-intercept is -1. Since both lines have the same slope, they are parallel, even though they cross the y-axis at different points.

The ability to quickly convert linear equations into slope-intercept form is a valuable skill for solving problems related to parallel and perpendicular lines. If an equation is given in a different form, such as standard form (Ax + By = C), you can rearrange it algebraically to isolate y and obtain the slope-intercept form. This involves performing operations like adding or subtracting terms from both sides and dividing by the coefficient of y. Once the equation is in slope-intercept form, identifying the slope becomes straightforward, and you can readily compare it with the slopes of other lines to determine parallelism or other relationships. Mastering the slope-intercept form is therefore a fundamental step in understanding and manipulating linear equations.

Deconstructing the Problem: Identifying the Slope

The given problem states that line k is parallel to the line with the equation y = 7x + 5. Our primary objective is to identify which of the provided options could represent the equation of line k. To achieve this, we must first extract the slope of the given line. As discussed earlier, the equation y = 7x + 5 is already in slope-intercept form (y = mx + b), making the task of identifying the slope remarkably simple. By comparing the given equation with the slope-intercept form, we can directly see that the coefficient of x, which represents the slope m, is 7. Therefore, the slope of the line y = 7x + 5 is 7.

Since line k is parallel to this line, it must have the same slope. This is the crucial piece of information that will guide us in selecting the correct answer from the options provided. We now know that any equation representing line k must also have a slope of 7. The y-intercept of line k can be different from 5, as parallel lines only need to have the same slope, not the same y-intercept. The y-intercept determines where the line crosses the y-axis, but it does not affect the line's direction or steepness, which are solely determined by the slope. Therefore, to solve this problem, we need to examine the given options and identify the equation that has a slope of 7, regardless of its y-intercept. This understanding simplifies the problem significantly, allowing us to focus on the most critical aspect: the slope.

Evaluating the Options: Finding the Match

Now that we know line k must have a slope of 7, let's examine the given options one by one:

A. y = -7x + 5

B. y = -1/7 x + 4

C. y = 5x + 1/5

D. y = 7x - 1/5

Each of these equations is presented in slope-intercept form, making it easy to identify their slopes. To determine which equation could represent line k, we simply need to find the one with a slope of 7.

Option A: y = -7x + 5. The slope in this equation is -7. This is the negative of the slope we are looking for, so this option is incorrect.

Option B: y = -1/7 x + 4. The slope in this equation is -1/7. This is a negative fraction, and it is clearly not equal to 7, so this option is also incorrect.

Option C: y = 5x + 1/5. The slope in this equation is 5. While this is a positive number, it is not equal to 7, so this option is incorrect as well.

Option D: y = 7x - 1/5. The slope in this equation is 7. This matches the slope of the given line, so this is the correct answer.

By systematically evaluating each option and comparing its slope to the required slope of 7, we can confidently identify option D as the correct equation for line k. This process highlights the importance of understanding the slope-intercept form and the relationship between slopes and parallel lines.

Conclusion: Mastering Linear Equations

In conclusion, the correct answer is D. y = 7x - 1/5. This problem effectively demonstrates the crucial connection between the slopes of parallel lines and their equations. By understanding that parallel lines have the same slope and by mastering the slope-intercept form of a linear equation, you can readily solve problems involving parallel lines in the xy-plane.

This exercise underscores the importance of a solid foundation in linear equations for success in mathematics. The ability to identify slopes, interpret equations, and apply geometric principles is essential for tackling a wide range of problems. By consistently practicing and reinforcing these concepts, you can build confidence and proficiency in algebra and beyond. The principles discussed here extend beyond simple parallelism, providing a framework for understanding perpendicular lines, systems of equations, and other advanced topics in mathematics.

In the xy-plane, line k is parallel to the line given by the equation y = 7x + 5. Which of the following equations could be the equation of line k?

Parallel Lines in xy-Plane How to Find Equations of Parallel Lines