Finding The Equation Of A Line Passing Through Two Points (5, 21) And (-5, -29)
In mathematics, determining the equation of a line is a fundamental concept in coordinate geometry. Given two points on a Cartesian plane, we can find the unique line that passes through them. This article will delve into the step-by-step process of finding the equation of a line, using the points (5, 21) and (-5, -29) as an example. We will explore the underlying concepts, the formulas involved, and the practical application of these principles. Understanding how to find the equation of a line is crucial not only in mathematics but also in various fields such as physics, engineering, and computer graphics. By mastering this skill, you can model and analyze linear relationships, predict trends, and solve real-world problems more effectively. The equation of a line provides a concise and powerful way to represent linear relationships, making it an indispensable tool in both theoretical and applied contexts. We will start by understanding the basic concepts of linear equations and then move towards applying these concepts to find the equation of the line passing through the given points. This will involve calculating the slope and then using the point-slope form to derive the equation. Let’s begin by laying the groundwork for this exploration.
Understanding the Basics of Linear Equations
Before diving into the specifics of our problem, it's essential to grasp the basics of linear equations. A linear equation represents a straight line on a graph. The most common form of a linear equation is the slope-intercept form, which is expressed as y = mx + b, where m represents the slope of the line and b represents the y-intercept. The slope indicates how steep the line is, while the y-intercept is the point where the line crosses the y-axis. Another important form is the point-slope form, given by y - y₁ = m(x - x₁), where (x₁, y₁) is a point on the line and m is the slope. This form is particularly useful when we have a point and the slope, which is the case in our problem. The slope (m) can be calculated using two points (x₁, y₁) and (x₂, y₂) on the line using the formula m = (y₂ - y₁) / (x₂ - x₁). Understanding these forms and the concepts behind them is crucial for solving problems related to linear equations. The beauty of linear equations lies in their simplicity and versatility. They are used to model a wide range of phenomena, from the motion of objects to the relationship between supply and demand in economics. By understanding the components of a linear equation, you gain the ability to interpret and predict these phenomena. The slope and y-intercept provide key insights into the behavior of the line, and therefore the relationship it represents. For instance, a positive slope indicates a direct relationship, while a negative slope indicates an inverse relationship. In the following sections, we will apply these fundamental concepts to find the equation of the line passing through the given points. Now, let's move on to the first step in solving our problem: calculating the slope.
Calculating the Slope
The slope of a line is a measure of its steepness and direction. It tells us how much the y-value changes for every unit change in the x-value. To find the slope (m) of a line passing through two points (x₁, y₁) and (x₂, y₂), we use the formula: m = (y₂ - y₁) / (x₂ - x₁). In our case, the points are (5, 21) and (-5, -29). Let's designate (5, 21) as (x₁, y₁) and (-5, -29) as (x₂, y₂). Plugging these values into the formula, we get: m = (-29 - 21) / (-5 - 5). Simplifying the numerator, we have -29 - 21 = -50. Simplifying the denominator, we have -5 - 5 = -10. Therefore, m = -50 / -10. Dividing -50 by -10, we find that m = 5. This means that for every 1 unit increase in x, the y-value increases by 5 units. The slope is a critical component of the equation of a line, as it determines the line's direction and steepness. A positive slope indicates that the line is increasing from left to right, while a negative slope indicates that the line is decreasing. A slope of 0 represents a horizontal line, and an undefined slope represents a vertical line. Now that we have calculated the slope, we can use this information along with the point-slope form to find the equation of the line. The point-slope form allows us to write the equation of a line using a single point and the slope, making it a powerful tool in coordinate geometry. In the next section, we will apply the point-slope form to our problem and derive the equation of the line. So, let's proceed to the next step and utilize the slope we just calculated.
Using the Point-Slope Form
Now that we have the slope, m = 5, 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 (x₁, y₁) is a point on the line and m is the slope. We have two points to choose from: (5, 21) and (-5, -29). Let's use the point (5, 21) as (x₁, y₁). Substituting the values into the point-slope form, we get: y - 21 = 5(x - 5). Next, we need to simplify this equation to the slope-intercept form, y = mx + b. First, distribute the 5 on the right side of the equation: y - 21 = 5x - 25. Now, add 21 to both sides of the equation to isolate y: y = 5x - 25 + 21. Simplifying further, we get: y = 5x - 4. This is the equation of the line in slope-intercept form. The point-slope form is a versatile tool for finding the equation of a line because it allows us to use any point on the line along with the slope. This flexibility is particularly useful when we are given two points and need to find the equation of the line. The ability to transform the equation from point-slope form to slope-intercept form is also crucial, as the slope-intercept form provides a clear understanding of the line's slope and y-intercept. This transformation involves algebraic manipulation, which is a fundamental skill in mathematics. In the next section, we will verify our equation by plugging in the coordinates of both points and ensuring that they satisfy the equation. This step is essential to confirm the accuracy of our solution. So, let's proceed to the verification process.
Verifying the Equation
To ensure that the equation y = 5x - 4 is correct, we need to verify that both points (5, 21) and (-5, -29) satisfy the equation. This involves substituting the x and y coordinates of each point into the equation and checking if the equation holds true. First, let's substitute the coordinates of the point (5, 21) into the equation: 21 = 5(5) - 4. Simplifying the right side, we get: 21 = 25 - 4, which simplifies to 21 = 21. This is true, so the point (5, 21) satisfies the equation. Next, let's substitute the coordinates of the point (-5, -29) into the equation: -29 = 5(-5) - 4. Simplifying the right side, we get: -29 = -25 - 4, which simplifies to -29 = -29. This is also true, so the point (-5, -29) satisfies the equation. Since both points satisfy the equation y = 5x - 4, we can confidently conclude that this is the correct equation of the line passing through the points (5, 21) and (-5, -29). Verification is a crucial step in solving mathematical problems, as it helps to identify any errors that may have occurred during the solution process. By plugging in the given values and ensuring that they satisfy the equation, we can increase our confidence in the accuracy of our solution. In this case, the verification process has confirmed that our equation is indeed correct. This final step completes our process of finding the equation of the line. In the following conclusion, we will summarize the steps we took and highlight the key concepts involved in solving this problem. So, let's proceed to the conclusion.
Conclusion
In this article, we have successfully found the equation of the line passing through the points (5, 21) and (-5, -29). The process involved several key steps, each building upon the previous one. First, we discussed the basics of linear equations, including the slope-intercept form (y = mx + b) and the point-slope form (y - y₁ = m(x - x₁)). We highlighted the importance of understanding the slope (m) and the y-intercept (b) in the context of linear relationships. Next, we calculated the slope of the line using the formula m = (y₂ - y₁) / (x₂ - x₁), which yielded a slope of 5. This step is crucial as the slope is a fundamental property of the line, indicating its steepness and direction. Then, we utilized the point-slope form with the calculated slope and one of the given points to derive the equation of the line. We chose the point (5, 21) and substituted the values into the point-slope form, which we then simplified to the slope-intercept form y = 5x - 4. This equation represents the line that passes through the given points. Finally, we verified our equation by substituting the coordinates of both points into the equation and confirming that they satisfied the equation. This step ensured the accuracy of our solution and provided confidence in our result. The equation y = 5x - 4 represents a linear relationship where the y-value increases by 5 for every unit increase in the x-value, and the line crosses the y-axis at -4. This comprehensive process demonstrates how to find the equation of a line given two points, a fundamental skill in mathematics with applications in various fields. By understanding the concepts and steps involved, you can confidently tackle similar problems and apply this knowledge to real-world scenarios. This concludes our exploration of finding the equation of a line, and we hope this detailed explanation has been helpful.
The equation of the line is:
