Equation Of A Line Parallel To Another Line Passing Through A Point

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Finding the equation of a line that satisfies specific conditions is a fundamental concept in coordinate geometry. In this article, we will explore how to determine the equation of a line that passes through a given point and is parallel to another given line. This problem combines the understanding of parallel lines, slopes, and point-slope form, which are essential tools in linear algebra and analytical geometry. By working through this problem, we will reinforce these concepts and demonstrate a practical application of these principles.

Understanding Parallel Lines and Slopes

Before we dive into the specifics, let's revisit the key concepts that underpin this problem. Parallel lines are lines in the same plane that never intersect. A crucial property of parallel lines is that they have the same slope. The slope of a line measures its steepness and direction, and it is mathematically represented as the change in the y-coordinate divided by the change in the x-coordinate between any two points on the line. The slope is often denoted by the letter 'm'.

The equation of a line can be expressed in several forms, but the most relevant for this problem is the slope-intercept form: y = mx + b, where 'm' is the slope and 'b' is the y-intercept (the point where the line crosses the y-axis). Another important form is the point-slope form: y - y1 = m(x - x1), where (x1, y1) is a point on the line and 'm' is the slope. The point-slope form is particularly useful when we have a point and the slope, which is precisely the scenario we encounter in the problem we're addressing.

Determining the Slope from a Given Equation

In our problem, we are given a line in the general form 2x + y - 10 = 0. To find the slope of this line, we need to convert it into slope-intercept form (y = mx + b). This involves isolating 'y' on one side of the equation. Let's perform this conversion:

2x + y - 10 = 0
y = -2x + 10

Now, we can easily identify the slope. By comparing this equation with the slope-intercept form y = mx + b, we see that the slope m is -2. This is the slope of the given line, and since we want to find a line parallel to this one, the line we are looking for will also have a slope of -2. This insight is the cornerstone of solving our problem.

Applying the Point-Slope Form

We now know the slope of the line we want to find (m = -2) and a point it passes through (5, -7). This is the perfect setup for using the point-slope form of a line equation, which is y - y1 = m(x - x1). Here, (x1, y1) is the given point, and 'm' is the slope. Let's plug in the values:

  • x1 = 5
  • y1 = -7
  • m = -2

So the equation becomes:

y - (-7) = -2(x - 5)
y + 7 = -2(x - 5)

This is the equation of the line in point-slope form. While this form is perfectly valid, it's often preferable to express the equation in slope-intercept form (y = mx + b) or general form (Ax + By + C = 0) for clarity and ease of comparison with other linear equations. Let's convert it to slope-intercept form first.

Converting to Slope-Intercept Form

To convert the equation y + 7 = -2(x - 5) to slope-intercept form, we need to isolate 'y' on one side. Here are the steps:

  1. Distribute the -2 on the right side:

    y + 7 = -2x + 10

  2. Subtract 7 from both sides:

    y = -2x + 10 - 7
y = -2x + 3

Now, we have the equation in slope-intercept form: y = -2x + 3. This form clearly shows the slope (-2) and the y-intercept (3). This means the line crosses the y-axis at the point (0, 3).

Converting to General Form

The general form of a linear equation is Ax + By + C = 0, where A, B, and C are constants. To convert our equation to this form, we need to rearrange the terms so that all terms are on one side of the equation and the equation is equal to zero. Starting from the slope-intercept form y = -2x + 3, we can proceed as follows:

  1. Add 2x to both sides:

    2x + y = 3

  2. Subtract 3 from both sides:

    2x + y - 3 = 0

Now, the equation is in general form: 2x + y - 3 = 0. In this form, A = 2, B = 1, and C = -3. The general form is particularly useful for certain types of linear algebra operations and for comparing different linear equations.

Summarizing the Solution

In summary, to find the equation of a line that passes through the point (5, -7) and is parallel to the line 2x + y - 10 = 0, we followed these steps:

  1. Find the slope of the given line: We converted 2x + y - 10 = 0 to slope-intercept form (y = -2x + 10) and identified the slope as -2.
  2. Use the same slope for the parallel line: Since parallel lines have the same slope, the line we are looking for also has a slope of -2.
  3. Apply the point-slope form: We used the point-slope form y - y1 = m(x - x1) with the point (5, -7) and the slope -2 to get the equation y + 7 = -2(x - 5).
  4. Convert to slope-intercept form (optional): We converted the point-slope form to slope-intercept form (y = -2x + 3) for clarity.
  5. Convert to general form (optional): We also converted the equation to general form (2x + y - 3 = 0).

Therefore, the equation of the line that passes through the point (5, -7) and is parallel to the line 2x + y - 10 = 0 is y = -2x + 3 in slope-intercept form and 2x + y - 3 = 0 in general form.

Conclusion

Finding the equation of a line parallel to another line through a given point is a classic problem that reinforces key concepts in coordinate geometry. The process involves understanding the properties of parallel lines, identifying slopes, and applying the point-slope form of a linear equation. By mastering these techniques, you can tackle a wide range of problems in linear algebra and analytical geometry. The ability to convert between different forms of linear equations (slope-intercept, point-slope, and general form) is also a valuable skill. This problem serves as an excellent example of how these concepts come together to solve a practical problem in mathematics. The techniques discussed here are not only applicable in academic settings but also in various real-world applications, such as engineering, physics, and computer graphics, where understanding linear relationships is crucial.