Finding The Equation Of A Line Parallel To 2x + 5y = 10

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Finding the equation of a line that is parallel to a given line and passes through a specific point is a common problem in algebra. This article will guide you through the steps to solve this problem, providing a detailed explanation and solution. We will explore the concepts of parallel lines, slopes, and the point-slope form of a linear equation. Let's consider the given problem: What is the equation of a line that is parallel to the line $2x + 5y = 10$ and passes through the point (-5, 1)? Check all that apply.

Understanding Parallel Lines and Slopes

When dealing with parallel lines, it's crucial to understand their fundamental property: they have the same slope. The slope of a line determines its steepness and direction. If two lines have the same slope, they will never intersect, making them parallel. The equation of a line is often represented in the slope-intercept form, which is $y = mx + b$, where m represents the slope and b represents the y-intercept (the point where the line crosses the y-axis). Before we can find the equation of a line parallel to $2x + 5y = 10$, we need to determine the slope of this line. To do this, we will convert the given equation into slope-intercept form. Starting with $2x + 5y = 10$, we need to isolate y on one side of the equation. First, subtract $2x$ from both sides: $5y = -2x + 10$. Next, divide both sides by 5 to solve for y: $y = - rac{2}{5}x + 2$. Now, the equation is in slope-intercept form, and we can easily identify the slope. The slope of the given line is $- rac{2}{5}$. Since parallel lines have the same slope, any line parallel to $2x + 5y = 10$ will also have a slope of $- rac{2}{5}$. This understanding is a critical first step in solving our problem.

Using the Point-Slope Form

Now that we know the slope of the parallel line, we need to find its specific equation. We know that the parallel line passes through the point (-5, 1). To find the equation of the line, we can use the point-slope form. The point-slope form of a linear equation is given by: $y - y_1 = m(x - x_1)$, where m is the slope of the line, and $(x_1, y_1)$ is a point on the line. In our case, we have the slope $m = - rac2}{5}$ and the point $(-5, 1)$, so $x_1 = -5$ and $y_1 = 1$. Substitute these values into the point-slope form $y - 1 = - rac{25}(x - (-5))$. Simplify the equation $y - 1 = - rac{25}(x + 5)$. Next, distribute the $- rac{2}{5}$ across the terms inside the parentheses $y - 1 = - rac{25}x - 2$. Now, isolate y by adding 1 to both sides of the equation $y = - rac{25}x - 2 + 1$. This simplifies to $y = - rac{2{5}x - 1$. Thus, the equation of the line parallel to $2x + 5y = 10$ and passing through the point (-5, 1) is $y = - rac{2}{5}x - 1$. This equation is in slope-intercept form, which is useful for identifying the slope and y-intercept. However, we should also consider other forms of the equation to match the given answer choices.

Converting to Standard Form

In addition to the slope-intercept form, linear equations can also be expressed in standard form, which is $Ax + By = C$, where A, B, and C are constants. It's often necessary to convert between these forms to match the required format in a problem. We found the equation of the line in slope-intercept form as $y = - rac2}{5}x - 1$. To convert this to standard form, we want to eliminate the fraction and rearrange the terms. First, multiply the entire equation by 5 to eliminate the fraction $5y = -2x - 5$. Next, add $2x$ to both sides of the equation to get the terms x and y on the same side: $2x + 5y = -5$. Now, the equation is in standard form. Comparing this to the slope-intercept form we derived earlier, we now have two forms of the equation for the line parallel to $2x + 5y = 10$ and passing through (-5, 1). The standard form equation is $2x + 5y = -5$, and the slope-intercept form equation is $y = - rac{2{5}x - 1$. Both of these forms represent the same line, but they highlight different properties of the line. The slope-intercept form clearly shows the slope and y-intercept, while the standard form is often used for comparing equations and solving systems of equations.

Checking the Answer Choices

Now that we have derived the equations in both slope-intercept and standard forms, we can check the given answer choices to see which ones match. The original problem provided the following answer choices:

A. $y = - rac{2}{5}x - 1$ B. $2x + 5y = -5$ C. $y = - rac{2}{5}x - 3$ D. $2x + 5y = 10$

Let's analyze each choice:

  • A. $y = - rac{2}{5}x - 1$: This equation matches the slope-intercept form we derived, so it is a correct answer.
  • B. $2x + 5y = -5$: This equation matches the standard form we derived, so it is also a correct answer.
  • **C. $y = - rac2}{5}x - 3$** This equation has the same slope as the given line, but the y-intercept is different. Let's check if this line passes through the point (-5, 1). Substitute $x = -5$ into the equation: $y = - rac{2{5}(-5) - 3 = 2 - 3 = -1$. Since the y-value is -1, not 1, this line does not pass through the point (-5, 1), so this answer is incorrect.
  • D. $2x + 5y = 10$: This is the original line given in the problem. While it has the same slope as the parallel line we are looking for, it is not the equation of the parallel line that passes through (-5, 1). Therefore, this answer is incorrect.

Therefore, the correct answers are A and B. The equation of the line that is parallel to $2x + 5y = 10$ and passes through the point (-5, 1) can be expressed as $y = - rac{2}{5}x - 1$ in slope-intercept form and as $2x + 5y = -5$ in standard form. These two equations are equivalent and represent the same line.

Key Steps for Finding Parallel Line Equations

To summarize, here are the key steps to find the equation of a line that is parallel to a given line and passes through a specific point:

  1. Find the slope of the given line: Convert the given equation to slope-intercept form ($y = mx + b$) to identify the slope m.
  2. Use the same slope for the parallel line: Parallel lines have the same slope, so the slope of the parallel line is also m.
  3. Use the point-slope form: Substitute the slope m and the given point $(x_1, y_1)$ into the point-slope form equation: $y - y_1 = m(x - x_1)$.
  4. Simplify the equation: Simplify the equation to the desired form (slope-intercept or standard form).
  5. Check the answer choices: Compare the derived equations with the given answer choices to identify the correct answers.

By following these steps, you can confidently solve problems involving parallel lines and their equations. Understanding the concepts of slope, point-slope form, and the different forms of linear equations is essential for success in algebra and beyond. Remember that practice makes perfect, so work through various examples to reinforce your understanding and skills. The ability to find the equation of a parallel line is a valuable skill in mathematics, with applications in various fields such as physics, engineering, and computer graphics.

Conclusion

In conclusion, finding the equation of a line parallel to a given line involves understanding the properties of parallel lines and the different forms of linear equations. The key is to recognize that parallel lines have the same slope and to use the point-slope form to derive the equation of the desired line. By converting the equation to slope-intercept or standard form, you can easily match the answer choices and verify your solution. In the given problem, the equations $y = - rac{2}{5}x - 1$ and $2x + 5y = -5$ both represent the line parallel to $2x + 5y = 10$ and passing through the point (-5, 1). This step-by-step approach ensures accuracy and a solid understanding of the underlying concepts.