Finding The Equation Of A Line Passing Through Points In A Table
How to write the equation of a line that passes through the points shown in the table?
In mathematics, determining the equation of a line is a fundamental concept. When provided with a set of points, the challenge lies in identifying the linear relationship that connects them. This article delves into the process of finding the equation of a line that passes through the points specified in a table. We will explore the methods and steps involved in deriving the equation, ensuring a clear and comprehensive understanding of the topic. This article will guide you through the process, providing a detailed explanation and practical steps to solve this type of problem.
Understanding Linear Equations
Before diving into the specifics, it's crucial to understand the basic forms of linear equations. The most common form is the slope-intercept form, which is expressed as y = mx + b. Here, 'm' represents the slope of the line, and 'b' is the y-intercept, the point where the line crosses the y-axis. The slope 'm' signifies the rate of change of 'y' with respect to 'x', indicating how much 'y' changes for every unit change in 'x'. Another useful form is the point-slope form, y - y1 = m(x - x1), where (x1, y1) is a known point on the line and 'm' is the slope. This form is particularly helpful when you have a point and the slope or can calculate the slope from two points. Understanding these forms is essential for finding the equation of a line from given points. The slope-intercept form provides a straightforward way to visualize the line’s behavior, while the point-slope form is useful for constructing the equation when you have a specific point and the slope. Grasping these concepts sets the stage for effectively tackling the problem at hand.
Calculating the Slope (m)
To find the equation of a line, the first critical step is to determine its slope, denoted by 'm'. The slope represents the steepness and direction of the line. Given two points on the line, (x1, y1) and (x2, y2), the slope can be calculated using the formula: m = (y2 - y1) / (x2 - x1). This formula essentially calculates the change in the y-coordinates divided by the change in the x-coordinates. Let's apply this to the points provided in the table: (-10, 2), (-4, 1), (8, -1), and (14, -2). We can pick any two points to calculate the slope. For instance, let's use the points (-10, 2) and (-4, 1). Plugging these values into the slope formula, we get m = (1 - 2) / (-4 - (-10)) = -1 / 6. We can verify this by using another pair of points, such as (8, -1) and (14, -2). Using these points, m = (-2 - (-1)) / (14 - 8) = -1 / 6. The slope remains consistent, which confirms that these points lie on a straight line. This consistency is a key indicator that we are dealing with a linear relationship. Calculating the slope accurately is crucial as it forms the foundation for determining the rest of the line's equation. A clear understanding of how to derive the slope ensures that the subsequent steps are built on a solid mathematical basis.
Finding the Y-Intercept (b)
Once we have calculated the slope 'm', the next step in determining the equation of the line in slope-intercept form (y = mx + b) is to find the y-intercept, denoted by 'b'. The y-intercept is the point where the line crosses the y-axis, which occurs when x = 0. To find 'b', we can use the slope we calculated and any point from the table. Let's use the point-slope form of the equation: y - y1 = m(x - x1). We already found that m = -1/6. Now, let's use the point (-10, 2). Plugging these values into the point-slope form, we get y - 2 = (-1/6)(x - (-10)). Simplifying this equation, we have y - 2 = (-1/6)(x + 10). Expanding the equation gives us y - 2 = (-1/6)x - (10/6). To isolate 'y' and get the equation in slope-intercept form, we add 2 to both sides: y = (-1/6)x - (10/6) + 2. Converting 2 to a fraction with a denominator of 6, we get 2 = 12/6. Thus, the equation becomes y = (-1/6)x - (10/6) + (12/6). Combining the constants, we find y = (-1/6)x + (2/6), which simplifies to y = (-1/6)x + (1/3). Therefore, the y-intercept 'b' is 1/3. This means the line crosses the y-axis at the point (0, 1/3). The y-intercept provides an essential anchor point for the line, allowing us to fully define its position in the coordinate plane. Accurately determining 'b' completes the process of defining the linear equation in slope-intercept form.
Constructing the Equation of the Line
With the slope 'm' and the y-intercept 'b' determined, we can now construct the equation of the line. We found that the slope m = -1/6 and the y-intercept b = 1/3. Using the slope-intercept form of the linear equation, which is y = mx + b, we can substitute these values directly into the equation. This gives us y = (-1/6)x + (1/3). This equation represents the line that passes through the given points in the table. To ensure accuracy, we can verify this equation by plugging in the x-values from the table and checking if the corresponding y-values match. For example, let's check the point (-10, 2). Substituting x = -10 into the equation, we get y = (-1/6)(-10) + (1/3) = (10/6) + (1/3). Converting 1/3 to 2/6 to have a common denominator, we have y = (10/6) + (2/6) = 12/6 = 2. This matches the y-value in the table for x = -10. Let's check another point, (8, -1). Substituting x = 8 into the equation, we get y = (-1/6)(8) + (1/3) = (-8/6) + (1/3). Converting 1/3 to 2/6, we have y = (-8/6) + (2/6) = -6/6 = -1. This also matches the y-value in the table for x = 8. These checks confirm that the equation y = (-1/6)x + (1/3) accurately represents the line passing through the given points. Constructing the equation in this way ensures that we have a clear and concise representation of the linear relationship.
Verifying the Equation
After constructing the equation of the line, it is essential to verify its accuracy. This step ensures that the equation correctly represents the relationship between the x and y values provided in the table. To verify the equation y = (-1/6)x + (1/3), we can substitute each x-value from the table into the equation and check if the resulting y-value matches the corresponding y-value in the table. Let’s start with the point (-4, 1). Substituting x = -4 into the equation, we get y = (-1/6)(-4) + (1/3) = (4/6) + (1/3). Converting 1/3 to 2/6 to have a common denominator, we have y = (4/6) + (2/6) = 6/6 = 1. This matches the y-value in the table for x = -4. Next, let's check the point (14, -2). Substituting x = 14 into the equation, we get y = (-1/6)(14) + (1/3) = (-14/6) + (1/3). Converting 1/3 to 2/6, we have y = (-14/6) + (2/6) = -12/6 = -2. This also matches the y-value in the table for x = 14. By verifying the equation with multiple points from the table, we can be confident in its accuracy. If the calculated y-values consistently match the given y-values, the equation is a reliable representation of the line. This step-by-step verification process is crucial in ensuring the equation accurately models the linear relationship between the points.
Conclusion
In summary, finding the equation of a line that passes through a set of given points involves several key steps. First, we calculate the slope 'm' using the formula m = (y2 - y1) / (x2 - x1). Then, we determine the y-intercept 'b' by using the slope and a point from the table in the point-slope form or directly in the slope-intercept form. Once we have both 'm' and 'b', we construct the equation of the line in the form y = mx + b. Finally, we verify the equation by substituting the x-values from the table and checking if the resulting y-values match the given y-values. This process ensures that the equation accurately represents the linear relationship between the points. By following these steps, one can confidently determine the equation of a line from a given set of points. Understanding linear equations is fundamental in mathematics and has practical applications in various fields. This comprehensive approach provides a clear and effective method for solving such problems, enhancing mathematical proficiency and problem-solving skills.