Finding The Equation Of A Line Through (3, 10) And (7, 28)

In the realm of mathematics, particularly in coordinate geometry, determining the equation of a straight line is a fundamental concept. A straight line can be uniquely defined by two points that lie on it. This article delves into the process of finding the equation of a straight line that passes through the points (3, 10) and (7, 28). The final equation will be expressed in the slope-intercept form, which is y = mx + c, where m represents the slope of the line and c represents the y-intercept. This form is widely used due to its simplicity and the direct insight it provides into the line's characteristics. Understanding how to derive this equation is crucial for various applications, including physics, engineering, and computer graphics. This exploration will not only provide the answer but also elucidate the underlying principles of linear equations, offering a comprehensive understanding for students and enthusiasts alike. The ability to find the equation of a line given two points is a cornerstone of analytical geometry, allowing us to model and analyze linear relationships effectively. This skill is not just theoretical; it has practical applications in fields ranging from data analysis to computer graphics, where lines are used to represent relationships and boundaries. This article will meticulously walk through the steps required to calculate the slope and y-intercept, ensuring that readers can confidently apply these methods to similar problems in the future. Mastering this process is a key step in developing a strong foundation in mathematics and its applications.

Determining the Slope (m)

The slope, represented by m, is a crucial parameter that defines the steepness and direction of a line. It is calculated using the formula:

m = (y2 - y1) / (x2 - x1)

where (x1, y1) and (x2, y2) are the coordinates of the two given points. In this case, our points are (3, 10) and (7, 28). Let's assign (3, 10) as (x1, y1) and (7, 28) as (x2, y2). Substituting these values into the slope formula, we get:

m = (28 - 10) / (7 - 3) = 18 / 4 = 9 / 2

Thus, the slope of the line is 9/2. This positive slope indicates that the line is increasing, meaning that as the x-value increases, the y-value also increases. The magnitude of the slope, 9/2, tells us that for every 2 units the line moves horizontally, it rises 9 units vertically. This steepness is a key characteristic of the line, influencing its position and behavior on the coordinate plane. A steeper line will have a larger slope magnitude, while a flatter line will have a smaller slope magnitude. Understanding the slope is essential for interpreting and working with linear equations, as it directly affects how the line is graphed and used in various applications. The calculation of the slope is a fundamental step in finding the equation of a line, as it provides the rate of change between the two variables. This rate of change is constant for a straight line, making it a linear relationship. The slope not only defines the direction and steepness of the line but also plays a critical role in determining the y-intercept, which is the point where the line crosses the y-axis. Therefore, accurately calculating the slope is the first key step in finding the complete equation of the line.

Finding the y-intercept (c)

Now that we have the slope m (9/2), we can proceed to find the y-intercept, represented by c. The y-intercept is the point where the line crosses the y-axis, which occurs when x = 0. To find c, we can use the slope-intercept form of the equation, y = mx + c, and substitute the coordinates of one of the given points along with the calculated slope. Let's use the point (3, 10). Substituting x = 3, y = 10, and m = 9/2 into the equation, we get:

10 = (9/2) * 3 + c

Now, we solve for c:

10 = 27/2 + c c = 10 - 27/2 c = 20/2 - 27/2 c = -7/2

Therefore, the y-intercept c is -7/2. This value indicates that the line intersects the y-axis at the point (0, -7/2). The y-intercept is a crucial component of the line's equation, as it determines the vertical position of the line on the coordinate plane. A positive y-intercept means the line crosses the y-axis above the x-axis, while a negative y-intercept means it crosses below the x-axis. The y-intercept, along with the slope, uniquely defines the position and orientation of the line. Finding the y-intercept is a necessary step in completing the equation of the line, providing the final piece of information needed to describe the line's behavior. The process of solving for c involves algebraic manipulation, ensuring that we isolate the y-intercept on one side of the equation. This step is critical for accurately representing the line in its slope-intercept form, which is widely used and easily understood. The y-intercept not only serves as a point of reference but also helps in graphing the line and understanding its relationship with other lines and points on the coordinate plane.

The Equation of the Line

With the slope (m = 9/2) and the y-intercept (c = -7/2) determined, we can now write the equation of the line in the form y = mx + c. Substituting the values we found, the equation of the line that passes through the points (3, 10) and (7, 28) is:

y = (9/2)x - 7/2

This equation represents the unique straight line that connects the two given points. It tells us that for every increase of 2 in the x-value, the y-value increases by 9, and the line crosses the y-axis at -7/2. The equation is in its simplest form, with the slope and y-intercept expressed as fractions. This form is not only concise but also provides a clear understanding of the line's characteristics. The slope-intercept form y = mx + c is a powerful tool for representing linear relationships, as it directly shows the rate of change (slope) and the starting point (y-intercept). The equation we derived is a complete representation of the line, allowing us to predict the y-value for any given x-value and vice versa. It also enables us to analyze the line's behavior, such as its direction and steepness. The ability to find the equation of a line given two points is a fundamental skill in algebra and geometry, with applications in various fields, including physics, engineering, and economics. This equation serves as a mathematical model of the line, capturing its essential properties and allowing us to work with it analytically. The process of finding this equation involves a combination of algebraic techniques and geometric understanding, highlighting the interconnectedness of mathematical concepts.

In conclusion, the equation of the straight line that passes through the points (3, 10) and (7, 28) is y = (9/2)x - 7/2. We arrived at this equation by first calculating the slope using the formula m = (y2 - y1) / (x2 - x1) and then finding the y-intercept by substituting the slope and one of the points into the slope-intercept form y = mx + c. This process demonstrates a fundamental concept in coordinate geometry, providing a clear and methodical approach to finding the equation of a line given two points. The slope-intercept form of the equation provides valuable insights into the line's characteristics, allowing us to understand its steepness, direction, and position on the coordinate plane. This skill is essential for various mathematical applications and real-world problem-solving scenarios. The ability to determine the equation of a line is a cornerstone of linear algebra and analytical geometry, with wide-ranging applications in fields such as engineering, physics, computer science, and economics. The steps outlined in this article provide a solid foundation for further exploration of linear relationships and their representations. Mastering this process not only enhances mathematical proficiency but also strengthens problem-solving skills and logical reasoning. The equation y = (9/2)x - 7/2 is a precise mathematical representation of the line, capturing its essence and allowing us to work with it analytically. This equation can be used to predict the y-value for any given x-value, graph the line accurately, and analyze its relationship with other lines and points on the coordinate plane. The process of finding this equation highlights the power of mathematical tools and their ability to model and describe real-world phenomena.