Equation Of A Parallel Line With A Given X-Intercept

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In mathematics, determining the equation of a line that satisfies certain conditions is a fundamental concept in coordinate geometry. This article delves into the process of finding the equation of a line that is parallel to a given line and possesses a specific x-intercept. We will explore the underlying principles, step-by-step methods, and provide illustrative examples to solidify your understanding. Whether you are a student grappling with linear equations or simply seeking to refresh your knowledge, this comprehensive guide will equip you with the necessary tools to tackle such problems with confidence.

Understanding Parallel Lines and Their Equations

Before we embark on solving the problem at hand, it's crucial to establish a firm grasp of the properties of parallel lines and their equations.

Parallel lines, in the realm of Euclidean geometry, are lines that lie in the same plane and never intersect, no matter how far they are extended. A key characteristic of parallel lines is that they possess the same slope. The slope of a line, often denoted by the letter 'm', quantifies its steepness or inclination with respect to the horizontal axis. A line with a positive slope rises as you move from left to right, while a line with a negative slope descends. Parallel lines share the same slope, indicating that they have the same rate of change in their vertical position for every unit change in their horizontal position.

The equation of a line can be expressed in several forms, but the most common and widely used form is the slope-intercept form. The slope-intercept form of a linear equation is given by:

y = mx + b

where:

  • y represents the vertical coordinate of a point on the line
  • x represents the horizontal coordinate of a point on the line
  • m represents the slope of the line
  • b represents the y-intercept of the line

The y-intercept is the point where the line intersects the vertical axis (y-axis). It is the value of y when x is equal to 0. The slope-intercept form provides a clear and concise way to represent the equation of a line, as it directly reveals the slope and y-intercept, two crucial parameters that define the line's orientation and position on the coordinate plane.

When dealing with parallel lines, the slope-intercept form becomes particularly useful. Since parallel lines have the same slope, their equations will have the same 'm' value. However, their y-intercepts ('b' values) will be different, ensuring that the lines do not coincide and remain distinct.

Determining the Equation of a Parallel Line

Now that we have established the foundational principles of parallel lines and their equations, let's delve into the process of finding the equation of a line that is parallel to a given line and possesses a specific x-intercept. The x-intercept is the point where the line intersects the horizontal axis (x-axis). It is the value of x when y is equal to 0.

To find the equation of a line parallel to a given line and with a specific x-intercept, we can follow these steps:

  1. Identify the slope of the given line: The slope of the given line can be readily identified from its equation if it is in slope-intercept form (y = mx + b). The coefficient of the x term, 'm', represents the slope of the line. If the equation is not in slope-intercept form, we need to rearrange it algebraically to isolate y on one side of the equation. For instance, if we have an equation in the form Ax + By = C, we can rearrange it as follows:

    By = -Ax + C

y = (-A/B)x + (C/B)

Now, the slope of the line is -A/B.

  1. Determine the slope of the parallel line: Since parallel lines have the same slope, the slope of the parallel line will be the same as the slope of the given line. This is a fundamental property of parallel lines that simplifies the process of finding their equations.

  2. Use the x-intercept to find the y-intercept: The x-intercept is the point where the line crosses the x-axis, meaning that the y-coordinate at this point is 0. Let the x-intercept be denoted as (xâ‚€, 0). We can substitute the slope (m) of the parallel line and the x-intercept (xâ‚€, 0) into the slope-intercept form (y = mx + b) and solve for the y-intercept (b). The substitution will yield:

    0 = m * xâ‚€ + b

    Solving for b, we get:

    b = -m * xâ‚€

  3. Write the equation of the parallel line: Now that we have determined both the slope (m) and the y-intercept (b) of the parallel line, we can write its equation in slope-intercept form (y = mx + b). Simply substitute the values of m and b into the equation, and we obtain the equation of the line that is parallel to the given line and has the specified x-intercept.

Illustrative Examples

To solidify your understanding of the process, let's work through a few examples.

Example 1:

Find the equation of the line that is parallel to the line y = 2x + 3 and has an x-intercept of -1.

  • Step 1: Identify the slope of the given line.

The given line is in slope-intercept form (y = mx + b), where m = 2. Therefore, the slope of the given line is 2.

  • Step 2: Determine the slope of the parallel line.

    Since parallel lines have the same slope, the slope of the parallel line is also 2.

  • Step 3: Use the x-intercept to find the y-intercept.

The x-intercept is given as (-1, 0). Substituting the slope (m = 2) and the x-intercept (-1, 0) into the slope-intercept form (y = mx + b), we get:

0 = 2 * (-1) + b

Solving for b, we get:

b = 2

  • Step 4: Write the equation of the parallel line.

    Now that we have the slope (m = 2) and the y-intercept (b = 2), we can write the equation of the parallel line in slope-intercept form:

y = 2x + 2

Therefore, the equation of the line that is parallel to y = 2x + 3 and has an x-intercept of -1 is y = 2x + 2.

Example 2:

Find the equation of the line that is parallel to the line 3x + 4y = 12 and has an x-intercept of 4.

  • Step 1: Identify the slope of the given line.

The given line is not in slope-intercept form. We need to rearrange it to isolate y:

4y = -3x + 12

y = (-3/4)x + 3

Now, the equation is in slope-intercept form, and we can see that the slope of the given line is -3/4.

  • Step 2: Determine the slope of the parallel line.

    Since parallel lines have the same slope, the slope of the parallel line is also -3/4.

  • Step 3: Use the x-intercept to find the y-intercept.

The x-intercept is given as (4, 0). Substituting the slope (m = -3/4) and the x-intercept (4, 0) into the slope-intercept form (y = mx + b), we get:

0 = (-3/4) * 4 + b

Solving for b, we get:

b = 3

  • Step 4: Write the equation of the parallel line.

    Now that we have the slope (m = -3/4) and the y-intercept (b = 3), we can write the equation of the parallel line in slope-intercept form:

y = (-3/4)x + 3

Therefore, the equation of the line that is parallel to 3x + 4y = 12 and has an x-intercept of 4 is y = (-3/4)x + 3.

Solving the Given Problem

Now, let's apply the steps we've outlined to solve the original problem. We are given the line:

y = (2/3)x + 3

and we need to find the equation of a line parallel to this line with an x-intercept of -3.

  • Step 1: Identify the slope of the given line.

The given line is in slope-intercept form, and the slope (m) is 2/3.

  • Step 2: Determine the slope of the parallel line.

    The slope of the parallel line will also be 2/3.

  • Step 3: Use the x-intercept to find the y-intercept.

The x-intercept is given as -3, which corresponds to the point (-3, 0). Substituting the slope (m = 2/3) and the x-intercept (-3, 0) into the slope-intercept form (y = mx + b), we get:

0 = (2/3) * (-3) + b

Simplifying, we have:

0 = -2 + b

Solving for b, we get:

b = 2

  • Step 4: Write the equation of the parallel line.

    Now that we have the slope (m = 2/3) and the y-intercept (b = 2), we can write the equation of the parallel line in slope-intercept form:

y = (2/3)x + 2

Therefore, the equation of the line that is parallel to y = (2/3)x + 3 and has an x-intercept of -3 is y = (2/3)x + 2.

Conclusion

In this article, we have explored the concept of finding the equation of a line that is parallel to a given line and possesses a specific x-intercept. We have delved into the properties of parallel lines, the slope-intercept form of linear equations, and the significance of x-intercepts. By following the step-by-step method outlined and working through illustrative examples, you can confidently tackle such problems in mathematics. Understanding these fundamental concepts of coordinate geometry is essential for further exploration in more advanced mathematical topics.

The key takeaway is that parallel lines share the same slope, and the x-intercept provides a crucial point for determining the y-intercept of the parallel line. By combining these principles with the slope-intercept form, we can efficiently derive the equation of the desired line. This skill is not only valuable in academic settings but also has practical applications in various fields, such as engineering, physics, and computer graphics.

Remember to practice these concepts with various examples to solidify your understanding and build your problem-solving skills. With a solid grasp of linear equations and their properties, you'll be well-equipped to tackle more complex mathematical challenges.