Finding The Equation Of A Line Passing Through Two Points
In the realm of mathematics, particularly in coordinate geometry, determining the equation of a line is a fundamental concept. Among the various forms of linear equations, the slope-intercept form stands out for its clarity and ease of use. This form, expressed as y = mx + b, where m represents the slope and b represents the y-intercept, provides a direct way to understand the line's characteristics. This article delves into a step-by-step approach to finding the equation of a line in slope-intercept form, using two given points as the basis for our calculations. Specifically, we will tackle the problem of finding the equation of the line that passes through the points (-3, -12) and (-8, 5). By understanding the underlying principles and applying the appropriate formulas, you'll be equipped to solve similar problems with confidence.
Understanding Slope-Intercept Form
Before diving into the problem, it's crucial to have a firm grasp of the slope-intercept form itself. As mentioned earlier, the slope-intercept form of a linear equation is y = mx + b. Here, m is the slope of the line, which indicates its steepness and direction. A positive slope means the line rises as you move from left to right, while a negative slope indicates the line falls. The y-intercept, denoted by b, is the point where the line intersects the y-axis. This is the value of y when x is equal to 0. The beauty of the slope-intercept form lies in its ability to provide a quick visual representation of the line. By simply looking at the equation, you can immediately identify the slope and the y-intercept, which are essential for graphing the line and understanding its behavior.
Knowing the slope and y-intercept allows us to easily visualize and graph the line. For instance, if we have the equation y = 2x + 3, we know the slope is 2 and the y-intercept is 3. This means for every 1 unit we move to the right on the x-axis, the line rises 2 units on the y-axis. The line also crosses the y-axis at the point (0, 3). This simple interpretation makes the slope-intercept form a powerful tool in linear algebra and coordinate geometry. Moreover, understanding this form is crucial for various applications in real-world scenarios, such as modeling linear relationships in physics, economics, and computer science. Therefore, mastering the process of converting different forms of linear equations into slope-intercept form is a valuable skill for students and professionals alike.
Step 1: Calculate the Slope (m)
The first step in finding the equation of the line is to determine its slope. The slope, often denoted by m, represents the rate of change of y with respect to x. In simpler terms, it tells us how much the y-value changes for every unit change in the x-value. When given two points, (x₁, y₁) and (x₂, y₂), the slope can be calculated using the following formula:
m = (y₂ - y₁) / (x₂ - x₁)
This formula essentially calculates the rise (change in y) over the run (change in x). It's crucial to maintain consistency in the order of subtraction; if you start with y₂ in the numerator, you must start with x₂ in the denominator. Now, let's apply this formula to our given points, (-3, -12) and (-8, 5). We can designate (-3, -12) as (x₁, y₁) and (-8, 5) as (x₂, y₂). Plugging these values into the formula, we get:
m = (5 - (-12)) / (-8 - (-3))
Simplifying the expression, we have:
m = (5 + 12) / (-8 + 3)
m = 17 / -5
Therefore, the slope of the line passing through the points (-3, -12) and (-8, 5) is -17/5. This negative slope indicates that the line is decreasing as we move from left to right. Understanding the sign of the slope is crucial as it provides a quick visual cue about the line's direction. A steeper slope (larger absolute value) implies a more rapid change in y for a given change in x. The calculated slope will be used in the next steps to determine the complete equation of the line.
Step 2: Use the Point-Slope Form
Now that we have calculated the slope, the next step is to utilize the point-slope form of a linear equation. The point-slope form is a versatile tool that allows us to write the equation of a line when we know its slope and a point it passes through. The point-slope form is given by:
**y - y₁ = m(x - x₁) **
where m is the slope and (x₁, y₁) is any point on the line. This form is derived from the definition of slope and is particularly useful when converting to other forms of linear equations, such as the slope-intercept form. We already know the slope, m = -17/5, and we have two points to choose from: (-3, -12) and (-8, 5). Let's use the point (-3, -12) as (x₁, y₁). Plugging these values into the point-slope form, we get:
y - (-12) = (-17/5)(x - (-3))
Simplifying the equation, we have:
y + 12 = (-17/5)(x + 3)
This is the equation of the line in point-slope form. While this form is perfectly valid, it is often more convenient to express the equation in slope-intercept form, which we will do in the next step. The point-slope form provides a direct way to represent a line based on its slope and a single point, making it a valuable intermediary step in finding the slope-intercept form. Using either of the given points would yield the same final equation in slope-intercept form, highlighting the flexibility of this method. The key is to correctly substitute the values into the formula and simplify the resulting equation.
Step 3: Convert to Slope-Intercept Form (y = mx + b)
The final step in finding the equation of the line is to convert the point-slope form into the slope-intercept form, y = mx + b. This form is particularly useful because it explicitly shows the slope (m) and the y-intercept (b) of the line. We start with the equation we derived in the previous step:
y + 12 = (-17/5)(x + 3)
To convert this to slope-intercept form, we need to isolate y on one side of the equation. First, we distribute the slope (-17/5) to the terms inside the parentheses:
y + 12 = (-17/5)x + (-17/5)(3)
Simplifying the multiplication, we get:
y + 12 = (-17/5)x - 51/5
Next, we subtract 12 from both sides of the equation to isolate y:
y = (-17/5)x - 51/5 - 12
To combine the constants, we need to express 12 as a fraction with a denominator of 5. 12 is equivalent to 60/5, so we have:
y = (-17/5)x - 51/5 - 60/5
Combining the fractions, we get:
y = (-17/5)x - 111/5
This is the equation of the line in slope-intercept form. We can see that the slope is -17/5, which we calculated earlier, and the y-intercept is -111/5. This means the line crosses the y-axis at the point (0, -111/5). Converting to slope-intercept form allows us to clearly see these key characteristics of the line and easily graph it or analyze its behavior. The process of distributing, simplifying, and isolating y is a fundamental technique in algebra and is essential for working with linear equations.
Final Answer
Therefore, the equation of the line in slope-intercept form that passes through the points (-3, -12) and (-8, 5) is:
y = (-17/5)x - 111/5
This equation represents a line with a negative slope, indicating it decreases from left to right, and intersects the y-axis at the point (0, -111/5). By following the steps outlined in this article – calculating the slope, using the point-slope form, and converting to slope-intercept form – you can confidently find the equation of any line given two points. Understanding these steps is crucial for mastering linear equations and their applications in various mathematical and real-world contexts. The slope-intercept form provides a clear and concise representation of a line, making it easy to analyze its properties and behavior. Mastering this process is a fundamental skill in algebra and serves as a building block for more advanced mathematical concepts.
