Finding The Equation Of A Line In Standard Form Given Two Points

In the realm of mathematics, particularly in coordinate geometry, determining the equation of a line is a fundamental concept. Understanding how to represent a line algebraically allows us to analyze its properties, such as its slope and intercepts, and to make predictions about its behavior. This article will guide you through the process of finding the equation of a line when given two points it passes through, and then express that equation in standard form. We'll tackle the specific problem of a line passing through the points (10, 5) and (13, -4), providing a detailed, step-by-step solution that you can apply to similar problems. This skill is crucial not only in mathematics but also in various real-world applications where linear relationships are used to model phenomena, such as in physics, economics, and computer graphics. By mastering this concept, you'll gain a powerful tool for problem-solving and analytical thinking. Understanding the equation of a line is crucial for several reasons. Firstly, it provides a concise way to describe the relationship between two variables, typically denoted as x and y. This relationship can be visualized as a straight line on a coordinate plane, where each point on the line represents a solution to the equation. Secondly, the equation of a line allows us to easily determine key properties of the line, such as its slope and y-intercept. The slope indicates the steepness and direction of the line, while the y-intercept tells us where the line crosses the vertical axis. These properties are essential for understanding the behavior of the line and for making predictions based on the linear relationship it represents. For instance, in economics, a linear equation might model the relationship between the price of a product and the quantity demanded. The slope of this line would indicate how much the quantity demanded changes for each unit change in price, and the y-intercept would represent the quantity demanded when the price is zero. Therefore, a solid grasp of how to find and interpret the equation of a line is invaluable for anyone working with quantitative data and linear models.

Problem Statement

Our specific challenge is to find the equation of a line that passes through two given points: (10, 5) and (13, -4). Furthermore, we need to express this equation in standard form, which is a specific format for linear equations. The standard form of a linear equation is typically written as Ax + By = C, where A, B, and C are constants, and A and B are not both zero. This form is particularly useful because it readily displays the coefficients of the x and y variables, which can be helpful for certain types of analysis and calculations. To solve this problem, we'll need to utilize the concept of slope, which measures the steepness of the line, and the point-slope form of a linear equation, which allows us to write the equation of a line given its slope and a point it passes through. Once we have the equation in point-slope form, we can manipulate it algebraically to arrive at the standard form. The ability to convert between different forms of linear equations is a key skill in algebra. For example, the slope-intercept form (y = mx + b) is useful for quickly identifying the slope (m) and y-intercept (b) of a line, while the standard form (Ax + By = C) is often preferred when dealing with systems of linear equations. Each form has its own advantages and is suited for different purposes. In this case, we are specifically asked to find the equation in standard form, so we will need to perform the necessary algebraic manipulations to transform our initial equation into this format. This involves rearranging terms, combining like terms, and ensuring that the coefficients meet the requirements of the standard form (usually, A is a positive integer). By mastering this process, you'll be able to confidently express linear equations in the form that best suits the problem you're trying to solve.

Step 1: Calculate the Slope

The first crucial step in finding the equation of the line is to determine its slope. The slope of a line, often denoted by the letter m, quantifies the line's steepness and direction. It represents the change in the y-coordinate (vertical change) for every unit change in the x-coordinate (horizontal change). A positive slope indicates that the line is increasing (going upwards) as you move from left to right, while a negative slope indicates that the line is decreasing (going downwards). A slope of zero means the line is horizontal, and an undefined slope means the line is vertical. To calculate the slope, we use the formula: m = (y₂ - y₁) / (x₂ - x₁), where (x₁, y₁) and (x₂, y₂) are the coordinates of two points on the line. In our case, the given points are (10, 5) and (13, -4). Let's designate (10, 5) as (x₁, y₁) and (13, -4) as (x₂, y₂). Plugging these values into the formula, we get: m = (-4 - 5) / (13 - 10). Simplifying the numerator, we have -4 - 5 = -9. Simplifying the denominator, we have 13 - 10 = 3. Therefore, the slope m is -9 / 3, which simplifies to -3. This means that for every one unit increase in x, the y value decreases by 3 units. Understanding the concept of slope is fundamental in linear algebra and has numerous applications in various fields. For example, in physics, the slope of a position-time graph represents the velocity of an object. In economics, the slope of a supply or demand curve indicates how the quantity supplied or demanded changes in response to a change in price. The negative slope we calculated here tells us that the line is decreasing, which could represent, for instance, a decreasing demand curve in economics, where as the price increases, the quantity demanded decreases. Therefore, accurately calculating and interpreting the slope is a critical first step in understanding and analyzing linear relationships.

Step 2: Use the Point-Slope Form

Now that we have calculated the slope (m = -3), the next step is to use the point-slope form of a linear equation to express the line's equation. The point-slope form is a versatile way to represent a linear equation, especially when we know the slope and a point that the line passes through. It is given by the formula: y - y₁ = m(x - x₁), where m is the slope, and (x₁, y₁) is any point on the line. We already know the slope, m = -3, and we have two points to choose from: (10, 5) and (13, -4). We can use either point; let's choose (10, 5) as our (x₁, y₁). Plugging these values into the point-slope form, we get: y - 5 = -3(x - 10). This equation represents the line in point-slope form. It tells us that the difference between y and 5 is equal to -3 times the difference between x and 10. This form is particularly useful because it directly incorporates the slope and a point on the line, making it easy to visualize the line's characteristics. The point-slope form highlights the relationship between the change in y and the change in x, which is the essence of the slope. By using this form, we can quickly write the equation of a line without having to first find the y-intercept. This is especially helpful when we are given two points, as in this problem. The point-slope form is also a crucial stepping stone to other forms of linear equations, such as the slope-intercept form and the standard form. It allows us to transition smoothly from the given information (slope and a point) to a more general representation of the line. In the next step, we will manipulate the point-slope equation we have obtained to express it in the desired standard form.

Step 3: Convert to Standard Form

Our final step is to convert the equation from point-slope form to standard form. As mentioned earlier, the standard form of a linear equation is Ax + By = C, where A, B, and C are constants, and A is typically a positive integer. To convert our equation, y - 5 = -3(x - 10), to standard form, we need to manipulate it algebraically to match this format. First, we distribute the -3 on the right side of the equation: y - 5 = -3x + 30. Next, we want to get the x and y terms on the same side of the equation. To do this, we add 3x to both sides: 3x + y - 5 = 30. Now, we want to isolate the constant term on the right side of the equation. We add 5 to both sides: 3x + y = 35. This is the equation of the line in standard form. We have successfully expressed the equation in the form Ax + By = C, where A = 3, B = 1, and C = 35. Converting to standard form is important for several reasons. It provides a consistent and easily recognizable format for linear equations, which simplifies comparisons and manipulations. It is also particularly useful when solving systems of linear equations, as the standard form makes it straightforward to use methods like elimination and substitution. Furthermore, the standard form highlights the coefficients of x and y, which can be relevant in certain applications. For example, in linear programming, the coefficients in the standard form often represent the costs or resources associated with the variables. The process of converting between different forms of linear equations is a fundamental skill in algebra. It demonstrates an understanding of algebraic manipulation and the properties of equality. By mastering this skill, you can confidently work with linear equations in any format and choose the form that is most convenient for the task at hand.

Solution

Therefore, the equation of the line that passes through the points (10, 5) and (13, -4), expressed in standard form, is 3x + y = 35. This corresponds to option D in the given choices. The solution we have found represents the unique linear relationship between x and y that satisfies the condition of passing through the two given points. It is a precise mathematical statement that encapsulates the line's behavior across the coordinate plane. The standard form of the equation, 3x + y = 35, provides a clear and concise representation of this relationship. We arrived at this solution by systematically applying the concepts of slope, point-slope form, and algebraic manipulation. Each step in the process was crucial: calculating the slope, using the point-slope form to write the equation, and then converting to standard form. This methodical approach ensures accuracy and allows us to break down complex problems into manageable steps. Understanding the process of solving such problems is as important as the solution itself. It equips you with the skills to tackle similar challenges and to apply these concepts in various contexts. For instance, you could use this method to find the equation of a line representing a trend in data, or to model a linear relationship in a scientific experiment. The ability to confidently determine the equation of a line is a valuable asset in mathematics and beyond. In conclusion, by following the steps outlined in this article, you can confidently find the equation of a line in standard form when given two points. This skill is a cornerstone of algebra and has wide-ranging applications in diverse fields. The correct answer to the problem is indeed option D, 3x + y = 35, but the journey to finding the solution is just as important as the answer itself.

Choosing the correct answer

Therefore, the correct answer is D. 3x + y = 35