Find The Equation Of A Parallel Line: A Step-by-Step Guide

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This question delves into the fundamental concepts of linear equations, parallel lines, and how to determine the equation of a line given specific conditions. To effectively solve this, we need to understand the relationship between the slopes of parallel lines and utilize the point-slope form of a linear equation. This article will provide a step-by-step explanation of how to solve this problem, ensuring you grasp the underlying principles and can apply them to similar scenarios.

Understanding Parallel Lines and Slopes

The cornerstone of solving this problem lies in understanding the characteristics of parallel lines. Parallel lines, by definition, are lines that never intersect. This crucial property translates directly to their slopes: parallel lines have the same slope. The slope of a line, often denoted by m, represents its steepness and direction. In the slope-intercept form of a linear equation, y = mx + b, the coefficient m directly represents the slope.

In the given equation, y = -1/3x + 4, we can readily identify the slope. Comparing it to the slope-intercept form, it's clear that the slope of this line is -1/3. Therefore, any line parallel to this one will also have a slope of -1/3. This understanding forms the basis for finding the equation of the parallel line we're seeking.

Furthermore, the y-intercept, represented by b in the slope-intercept form, dictates where the line crosses the y-axis. Parallel lines can have different y-intercepts, which is why simply knowing the slope isn't enough to define the parallel line uniquely. We need additional information, which in this case is the point (6, 5) that the line must pass through. This point provides the necessary constraint to determine the specific equation of the parallel line.

The process of finding the equation involves leveraging the known slope and the given point. We'll use the point-slope form of a linear equation, which is particularly useful when we have a point and a slope. This form allows us to construct the equation and then transform it into the slope-intercept form if desired. By understanding the significance of parallel lines having equal slopes and utilizing the point-slope form, we can systematically arrive at the solution.

Utilizing the Point-Slope Form

To determine the equation of the line, we will use the point-slope form of a linear equation. The point-slope form is given by:

y - y₁ = m(x - x₁)

where:

  • m is the slope of the line
  • (x₁, y₁) is a point on the line

We already know that the slope of the parallel line is -1/3 (the same as the given line). We also know that the line passes through the point (6, 5). We can now substitute these values into the point-slope form:

y - 5 = -1/3(x - 6)

This equation represents the line in point-slope form. To make it more readable and comparable to the answer choices, we need to simplify it and convert it to the slope-intercept form (y = mx + b).

First, distribute the -1/3 on the right side of the equation:

y - 5 = -1/3x + 2

Next, isolate y by adding 5 to both sides:

y = -1/3x + 2 + 5

y = -1/3x + 7

This is the equation of the line in slope-intercept form. It clearly shows the slope (-1/3) and the y-intercept (7). This form is particularly useful for visualizing the line and comparing it to other linear equations.

The point-slope form is a powerful tool in coordinate geometry, allowing us to construct the equation of a line when we have a point and the slope. It bridges the gap between graphical representation and algebraic expression, making it easier to analyze and manipulate linear equations. By understanding and applying the point-slope form, we can solve a wide range of problems related to lines and their properties.

Comparing with Answer Choices and Conclusion

Now that we have derived the equation of the line (y = -1/3x + 7), we can compare it with the given answer choices to find the correct one. The answer choices were:

A. y = -1/3x + 3 B. y = -1/3x + 7 C. y = 3x - 13 D. y = 3x + 5

By direct comparison, we can see that our derived equation y = -1/3x + 7 matches answer choice B. This confirms that our step-by-step solution was accurate and that we correctly applied the concepts of parallel lines and the point-slope form.

The other answer choices can be eliminated because they do not satisfy the conditions of the problem. Answer choice A has the correct slope but a different y-intercept, meaning it would be a parallel line but not pass through the point (6, 5). Answer choices C and D have slopes that are the negative reciprocal of -1/3, indicating that these lines are perpendicular to the given line, not parallel.

In conclusion, the equation of the line that is parallel to y = -1/3x + 4 and passes through the point (6, 5) is y = -1/3x + 7. This problem highlights the importance of understanding the relationship between the slopes of parallel lines and the application of the point-slope form in determining the equation of a line. By mastering these concepts, you can confidently tackle similar problems in coordinate geometry and linear algebra.

What is the equation of a line parallel to the line y = -1/3x + 4 that passes through the point (6, 5)?

Find the Equation of a Parallel Line A Step-by-Step Guide