Finding The Equation Of A Line Perpendicular To 3x + 5y = -9 And Passing Through (3, 0)
In the realm of coordinate geometry, a fundamental concept involves determining the equation of a line that is perpendicular to a given line and passes through a specific point. This problem combines the principles of linear equations, slopes, and the point-slope form of a line, offering a valuable exercise in analytical thinking. Let's delve into a comprehensive exploration of this concept, breaking down the steps involved and illustrating them with a concrete example.
Understanding the Fundamentals
Before we embark on the solution, it's crucial to grasp the underlying concepts:
- Linear Equations: A linear equation represents a straight line on a coordinate plane. It can be expressed in various forms, such as slope-intercept form (y = mx + b), standard form (Ax + By = C), and point-slope form (y - y1 = m(x - x1)).
- Slope: The slope of a line measures its steepness and direction. It is defined as the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line. A positive slope indicates an upward slant, while a negative slope indicates a downward slant. A zero slope represents a horizontal line, and an undefined slope represents a vertical line.
- Perpendicular Lines: Two lines are perpendicular if they intersect at a right angle (90 degrees). The slopes of perpendicular lines have a special relationship: they are negative reciprocals of each other. If the slope of one line is m, the slope of a line perpendicular to it is -1/m.
- Point-Slope Form: The point-slope form of a linear equation is a convenient way to represent a line when we know a point on the line (x1, y1) and its slope m. The equation is given by: y - y1 = m(x - x1).
Step-by-Step Solution
Let's consider the problem of finding the equation of a line that is perpendicular to the line 3x + 5y = -9 and passes through the point (3, 0). We can approach this problem in a systematic manner, following these steps:
1. Determine the Slope of the Given Line
The first step is to find the slope of the given line, 3x + 5y = -9. To do this, we can rewrite the equation in slope-intercept form (y = mx + b), where m represents the slope.
Subtracting 3x from both sides of the equation, we get:
5y = -3x - 9
Dividing both sides by 5, we obtain:
y = (-3/5)x - 9/5
From this equation, we can see that the slope of the given line is -3/5.
Understanding the slope is crucial for determining the equation of a perpendicular line. The slope, often denoted as m, signifies the steepness and direction of a line. In the equation y = mx + b, m is the slope. To find the slope of the given line, we need to convert the equation into slope-intercept form. This involves isolating y on one side of the equation. Through algebraic manipulation, we find the slope of the given line to be -3/5. This value is pivotal because the slope of any line perpendicular to it will be the negative reciprocal of -3/5.
2. Calculate the Slope of the Perpendicular Line
As mentioned earlier, the slopes of perpendicular lines are negative reciprocals of each other. Therefore, to find the slope of the line perpendicular to the given line, we take the negative reciprocal of -3/5.
The negative reciprocal of -3/5 is 5/3. This means that the slope of the perpendicular line is 5/3.
The concept of negative reciprocals is fundamental when dealing with perpendicular lines. Two lines are perpendicular if they intersect at a right angle (90 degrees), and their slopes are negative reciprocals of each other. If one line has a slope of m, the slope of a line perpendicular to it is -1/m. In our case, the slope of the given line is -3/5. Therefore, the slope of the line perpendicular to it is the negative reciprocal of -3/5, which is 5/3. This value, 5/3, is the slope we will use to define our new line. Understanding this relationship is critical for solving problems involving perpendicular lines.
3. Use the Point-Slope Form to Find the Equation
Now that we have the slope of the perpendicular line (5/3) and a point it passes through (3, 0), we can use the point-slope form of a linear equation to find the equation of the line.
The point-slope form is given by:
y - y1 = m(x - x1)
where m is the slope and (x1, y1) is the given point.
Substituting m = 5/3 and (x1, y1) = (3, 0) into the point-slope form, we get:
y - 0 = (5/3)(x - 3)
Simplifying, we have:
y = (5/3)x - 5
The point-slope form is a powerful tool for determining the equation of a line when we know a point on the line and its slope. The general form of the point-slope equation is y - y1 = m(x - x1), where m is the slope and (x1, y1) is the known point. In our scenario, we have the slope of the perpendicular line as 5/3 and the point (3, 0) through which it passes. Substituting these values into the point-slope form, we get y - 0 = (5/3)(x - 3). This equation is a direct representation of the line we are trying to find. The next step involves simplifying this equation to obtain a more standard form.
4. Convert to Standard Form (Optional)
While the equation y = (5/3)x - 5 is a valid representation of the line, we can also convert it to standard form (Ax + By = C) if desired.
To do this, we first multiply both sides of the equation by 3 to eliminate the fraction:
3y = 5x - 15
Then, we rearrange the terms to get the standard form:
5x - 3y = 15
Converting the equation to standard form can sometimes be necessary depending on the specific requirements of the problem or the desired format of the solution. The standard form of a linear equation is Ax + By = C, where A, B, and C are constants. To convert our equation y = (5/3)x - 5 to standard form, we first eliminate the fraction by multiplying the entire equation by 3. This gives us 3y = 5x - 15. Next, we rearrange the terms to match the standard form, which results in 5x - 3y = 15. This standard form provides another way to represent the equation of the line, and it highlights the relationship between x and y in a different way compared to the slope-intercept form.
Conclusion
Therefore, the equation of the line that is perpendicular to the line 3x + 5y = -9 and passes through the point (3, 0) is 5x - 3y = 15. This solution demonstrates the application of fundamental concepts in coordinate geometry, including linear equations, slopes, perpendicular lines, and the point-slope form. By understanding these concepts and following a systematic approach, you can confidently tackle similar problems involving perpendicular lines and their equations.
In summary, finding the equation of a line perpendicular to a given line and passing through a specific point involves a series of well-defined steps. First, we determine the slope of the given line by converting its equation to slope-intercept form. Second, we calculate the slope of the perpendicular line by taking the negative reciprocal of the given line's slope. Third, we use the point-slope form of a linear equation to construct the equation of the perpendicular line, utilizing the calculated slope and the given point. Finally, we can convert the equation to standard form if desired. This methodical approach ensures accuracy and clarity in solving these types of problems. Understanding each step and the underlying concepts allows for a deeper comprehension of coordinate geometry and its applications.
This detailed explanation and step-by-step solution should provide a comprehensive understanding of how to find the equation of a perpendicular line passing through a given point. Remember to practice similar problems to reinforce your understanding and develop your problem-solving skills in coordinate geometry.
Additional Practice
To further solidify your understanding, try solving similar problems with different given lines and points. For instance, consider the following:
- Find the equation of the line perpendicular to 2x - y = 4 and passing through the point (-1, 2).
- Determine the equation of the line perpendicular to y = -3x + 1 and passing through the point (0, -5).
- What is the equation of the line perpendicular to x + 4y = 7 and passing through the point (3, -2)?
Working through these practice problems will enhance your ability to apply the concepts and techniques discussed in this article. Remember to follow the steps outlined above, and don't hesitate to review the explanations if needed. With consistent practice, you'll become proficient in solving problems involving perpendicular lines and their equations.
The correct answer is 5x - 3y = 15.
