Finding Equation Of A Line With Slope 2/3 Through Point (5,8)

by ADMIN 62 views

In mathematics, determining the equation of a line is a fundamental concept with wide-ranging applications. Whether you're analyzing data, modeling real-world phenomena, or simply exploring geometric relationships, the ability to find the equation of a line is an essential skill. This article will delve into the process of finding the equation of a line when you're given its slope and a point it passes through. We'll explore the underlying principles, discuss different forms of linear equations, and provide step-by-step instructions with examples to solidify your understanding.

Understanding Slope and Linear Equations

Before we dive into the process of finding the equation, it's important to have a solid grasp of the key concepts involved:

  • Slope: The slope of a line, often denoted by the letter m, quantifies its steepness and direction. It represents the rate of change in the vertical direction (the rise) for every unit change in the horizontal direction (the run). A positive slope indicates an upward trend, while a negative slope indicates a downward trend. A slope of zero represents a horizontal line, and an undefined slope represents a vertical line.
  • Linear Equations: Linear equations are algebraic expressions that describe the relationship between two variables, typically x and y, and whose graph forms a straight line. There are several forms of linear equations, each with its own advantages and applications. The most common forms are:
    • Slope-intercept form: This form is expressed as y = mx + b, where m is the slope and b is the y-intercept (the point where the line crosses the y-axis). This form is particularly useful for quickly identifying the slope and y-intercept of a line.
    • Point-slope form: This form is expressed as y - y1 = m( x - x1), where m is the slope and (x1, y1) is a known point on the line. This form is highly convenient when you're given a point and the slope, as it directly incorporates this information into the equation.
    • Standard form: This form is expressed as Ax + By = C, where A, B, and C are constants. While not as directly informative as the other forms, the standard form is useful for certain algebraic manipulations and comparisons.

The Point-Slope Form: A Powerful Tool

The point-slope form of a linear equation is our primary tool for finding the equation of a line when given a slope and a point. Its structure, y - y1 = m( x - x1), elegantly incorporates the given information. The slope, m, directly represents the line's steepness, and the point (x1, y1) anchors the line in the coordinate plane. This form provides a straightforward pathway to construct the equation of the line, making it an indispensable tool in linear algebra and coordinate geometry. Understanding how to effectively utilize the point-slope form is crucial for solving a wide range of problems involving linear equations. It allows us to move from geometric information (a point and a slope) to an algebraic representation (the equation of the line), which can then be further analyzed or manipulated as needed. For instance, we can easily convert the point-slope form to the more familiar slope-intercept form or the standard form, depending on the specific requirements of the problem. The point-slope form serves as a bridge between the geometric and algebraic aspects of linear equations, empowering us to solve problems with greater efficiency and insight. Its ability to directly incorporate given information makes it a valuable asset in various mathematical contexts, from basic algebra to more advanced calculus and linear algebra applications.

Step-by-Step Guide: Finding the Equation

Now, let's outline the steps involved in finding the equation of a line using the point-slope form:

  1. Identify the given slope (m) and point (x1, y1). This is the information you'll use to construct the equation. The slope, m, is a numerical value representing the line's inclination, and the point (x1, y1) is a specific location on the line. Carefully note these values as they form the foundation of your calculation. Accurate identification of these values is crucial for obtaining the correct equation. A misidentification at this stage can lead to a completely different line, so take your time and double-check your work. The point-slope form is specifically designed to utilize this information, so ensuring you have the correct m, x1, and y1 is paramount to successfully applying the formula. This initial step sets the stage for the subsequent steps, where you'll substitute these values into the point-slope equation and simplify to obtain the line's equation in a more usable form, such as slope-intercept or standard form. Therefore, meticulous attention to detail in identifying the slope and point is essential for achieving an accurate result.
  2. Substitute the values of m, x1, and y1 into the point-slope form: y - y1 = m( x - x1). This is the core of the process, where you translate the given information into an algebraic expression. Replace the variables m, x1, and y1 with their respective numerical values. This substitution creates an equation specific to the line you're trying to define. The point-slope form acts as a template, and the substitution process fills in the details that uniquely characterize the line. It's crucial to substitute the values correctly, paying close attention to the signs. A negative slope or a negative coordinate can easily be mishandled, leading to an incorrect equation. After the substitution, you'll have an equation that represents the line in point-slope form, which directly reflects the given slope and point. This equation can then be further simplified or transformed into other forms, such as slope-intercept or standard form, depending on the specific requirements of the problem or the desired representation of the line.
  3. Simplify the equation (if necessary) to slope-intercept form (y = mx + b) or standard form (Ax + By = C). Once you've substituted the values into the point-slope form, you might want to rearrange the equation into a more familiar or useful form. This is where simplification comes in. You can distribute the slope (m) on the right side of the equation and then isolate y on the left side to obtain the slope-intercept form (y = mx + b). This form readily reveals the slope (m) and the y-intercept (b) of the line, making it convenient for graphing and understanding the line's behavior. Alternatively, you can rearrange the equation to the standard form (Ax + By = C), which is useful for certain algebraic manipulations and comparisons between lines. The choice of which form to simplify to often depends on the context of the problem or the specific information you need to extract from the equation. Regardless of the form you choose, the simplification process involves applying basic algebraic principles, such as distribution, addition, and subtraction, to manipulate the equation while maintaining its equality. This step transforms the equation from its initial point-slope form into a more readily interpretable or usable form.

Example: Finding the Equation

Let's illustrate this process with an example. Suppose we want to find the equation of a line that has a slope of m = 2/3 and passes through the point (5, 8). Here's how we can do it:

  1. Identify the given slope and point:
    • m = 2/3
    • (x1, y1) = (5, 8)
  2. Substitute the values into the point-slope form:
    • y - 8 = (2/3)(x - 5)
  3. Simplify the equation to slope-intercept form:
    • y - 8 = (2/3)x - 10/3
    • y = (2/3)x - 10/3 + 8
    • y = (2/3)x + 14/3

Therefore, the equation of the line in slope-intercept form is y = (2/3)x + 14/3.

Alternative Simplification to Standard Form

Alternatively, we could simplify the equation to standard form:

  1. Starting from the point-slope form: y - 8 = (2/3)(x - 5)
  2. Multiply both sides by 3 to eliminate the fraction: 3(y - 8) = 2(x - 5)
  3. Distribute: 3y - 24 = 2x - 10
  4. Rearrange to standard form: -2x + 3y = 14

Therefore, the equation of the line in standard form is -2x + 3y = 14.

Deep Dive into the Example: Unpacking the Solution

Let's dissect the example step-by-step to ensure a thorough understanding. We began with a slope of 2/3, indicating a gentle upward incline: for every 3 units we move horizontally, the line rises 2 units vertically. The point (5, 8) anchors this line in the coordinate plane, providing a specific location it must pass through. Substituting these values into the point-slope form, y - 8 = (2/3)(x - 5), creates an equation that embodies both the slope and the point. This equation, in its initial form, already defines the line, but it's often beneficial to simplify it into a more recognizable form. Simplifying to slope-intercept form, y = (2/3)x + 14/3, unveils the y-intercept (14/3), which is the point where the line crosses the vertical axis. This form is particularly useful for graphing the line, as we now know its slope and a specific point (the y-intercept). The alternative simplification to standard form, -2x + 3y = 14, showcases a different perspective on the same line. While the slope and y-intercept aren't immediately apparent in this form, it's valuable for comparing lines and performing certain algebraic manipulations. The process of converting between these forms highlights the flexibility of linear equations and the interconnectedness of different representations. Ultimately, each form provides a unique lens through which to view the line, and understanding how to navigate between them enhances our problem-solving capabilities in mathematics and beyond. This example underscores the power of the point-slope form as a starting point and demonstrates the versatility of linear equations in representing geometric relationships.

Conclusion

Finding the equation of a line given its slope and a point is a fundamental skill in mathematics. By understanding the point-slope form and following the steps outlined in this article, you can confidently determine the equation of any line given this information. Remember to practice with various examples to solidify your understanding and master this essential concept. The ability to manipulate linear equations is a cornerstone of mathematical literacy, enabling you to model real-world scenarios, analyze data, and solve a wide array of problems. The point-slope form, in particular, provides a direct link between geometric properties (slope and a point) and algebraic representation (the equation of the line), making it an invaluable tool in your mathematical toolkit. By mastering this concept, you'll not only enhance your mathematical proficiency but also gain a deeper appreciation for the elegance and power of linear relationships.