Finding Equation Of A Line Parallel To Another Line

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When dealing with linear equations in mathematics, one common task is to determine the equation of a line that satisfies certain conditions. One such condition is that the line must pass through a specific point and be parallel to another given line. This article will delve into the process of finding the equation of a line that meets these criteria, providing a step-by-step explanation and illustrative examples.

Understanding the Basics

Before we dive into the solution, it's essential to understand the fundamental concepts involved. A linear equation typically takes the form y = mx + b, where m represents the slope of the line and b represents the y-intercept. The slope indicates the steepness and direction of the line, while the y-intercept is the point where the line crosses the vertical axis. Parallel lines are lines that have the same slope but different y-intercepts. This means they run in the same direction but never intersect. To find the equation of a line, we need to determine its slope and y-intercept. When we are given a point that the line passes through and the equation of a line that it is parallel to, we can use this information to find the unknown equation. The given point will help us determine the y-intercept, while the slope of the parallel line will be the same as the slope of the line we are trying to find.

Determining the Slope

The first step in finding the equation of the desired line is to determine its slope. Since the line is parallel to the given line, it will have the same slope. If the equation of the given line is in slope-intercept form (y = mx + b), the slope is simply the coefficient of x. For example, if the given line has the equation y = (3/4)x + 1, then the slope of the given line is 3/4. Because parallel lines have the same slope, the line we want to find also has a slope of 3/4. If the equation of the given line is not in slope-intercept form, it may be necessary to rearrange the equation into slope-intercept form to easily identify the slope. Understanding the relationship between slopes of parallel lines is crucial for solving this type of problem, as it provides a direct link between the known line and the unknown line we are trying to define.

Using the Point-Slope Form

Now that we know the slope of the desired line, we can use the point-slope form of a linear equation to find its equation. The point-slope form is given by y - y₁ = m(x - x₁), where m is the slope of the line and (x₁, y₁) is a point that the line passes through. In our case, we are given the point (8, -5), so we can substitute these values into the point-slope form. We also know that the slope m is 3/4 (because the desired line is parallel to the line y = (3/4)x + 1). Substituting these values, we get y - (-5) = (3/4)(x - 8). This equation represents the line that passes through the point (8, -5) and has a slope of 3/4. However, it is not yet in the slope-intercept form, which is more commonly used. The point-slope form is a valuable tool because it allows us to construct the equation of a line directly from a known point and slope, without needing to explicitly calculate the y-intercept in the initial steps.

Converting to Slope-Intercept Form

The next step is to convert the equation from point-slope form to slope-intercept form (y = mx + b). To do this, we need to simplify the equation and isolate y. Starting with the equation y - (-5) = (3/4)(x - 8), we first simplify the left side to y + 5 = (3/4)(x - 8). Next, we distribute the 3/4 on the right side of the equation: y + 5 = (3/4)x - (3/4)(8). Simplifying further, we get y + 5 = (3/4)x - 6. Finally, we subtract 5 from both sides of the equation to isolate y: y = (3/4)x - 6 - 5. This gives us the equation y = (3/4)x - 11. This is the slope-intercept form of the equation, where the slope is 3/4 and the y-intercept is -11. The conversion to slope-intercept form is important because it clearly shows the slope and y-intercept of the line, making it easier to graph and compare with other lines. It also provides a standard form that is widely recognized and used in various mathematical contexts.

Verifying the Solution

To ensure that our solution is correct, we can verify that the line y = (3/4)x - 11 indeed passes through the point (8, -5) and has a slope of 3/4. To check if the line passes through the point (8, -5), we substitute x = 8 and y = -5 into the equation: -5 = (3/4)(8) - 11. Simplifying the right side, we get -5 = 6 - 11, which simplifies to -5 = -5. This confirms that the point (8, -5) lies on the line. We already know that the slope of the line is 3/4, as it is the coefficient of x in the slope-intercept form. Since this slope is the same as the slope of the given parallel line, our solution is consistent with the initial conditions. Verifying the solution is a crucial step in problem-solving, as it helps to catch any errors that may have occurred during the process and ensures that the final answer is accurate and reliable.

Conclusion

In summary, finding the equation of a line that passes through a given point and is parallel to another line involves several key steps. First, determine the slope of the given line. Since parallel lines have the same slope, this will be the slope of the desired line as well. Next, use the point-slope form of a linear equation to create an equation for the line using the given point and the determined slope. Finally, convert the equation to slope-intercept form to obtain the equation in the standard y = mx + b format. By following these steps, you can confidently find the equation of a line that meets the specified criteria. This process demonstrates a fundamental application of linear equations and their properties, highlighting the importance of understanding slopes, intercepts, and the relationships between parallel lines in solving geometric problems.

Therefore, the equation of the line that passes through (8, -5) and is parallel to the graphed line y = (3/4)x + 1 is y = (3/4)x - 11.

Practice Problems

To further solidify your understanding of this concept, try solving the following practice problems:

  1. Find the equation of the line that passes through the point (2, 3) and is parallel to the line y = 2x - 1.
  2. What is the equation of the line passing through (-1, 4) and parallel to y = -x + 5?
  3. Determine the equation of the line that passes through (0, -2) and is parallel to y = (1/2)x + 3.

Working through these problems will help you become more comfortable with the process and build your problem-solving skills in linear algebra.