Equation Of A Line Passing Through (-5,-1) And (1,-5)
In the realm of coordinate geometry, one of the fundamental tasks is to determine the equation of a line given certain information. A common scenario involves finding the equation of a line that passes through two distinct points. This article delves into the process of finding the equation of a line passing through the points (-5, -1) and (1, -5), providing a step-by-step guide and explanations to enhance understanding.
Understanding the Fundamentals
Before diving into the calculations, let's establish a firm grasp of the underlying concepts. A linear equation represents a straight line on a coordinate plane, and it can be expressed in various forms, including slope-intercept form, point-slope form, and standard form. Each form offers unique advantages depending on the given information and the desired representation of the equation.
The slope-intercept form is written as y = mx + b, where 'm' represents the slope of the line and 'b' represents the y-intercept (the point where the line crosses the y-axis). The slope (m) quantifies the steepness and direction of the line, calculated as the change in y divided by the change in x between two points on the line. The point-slope form is expressed as y - y₁ = m(x - x₁), where 'm' is the slope and (x₁, y₁) is a known point on the line. This form is particularly useful when the slope and a point on the line are known.
To determine the equation of a line passing through two points, we first calculate the slope using the coordinates of the given points. Then, we can utilize either the point-slope form or the slope-intercept form to derive the equation. Let's proceed with the step-by-step solution for the given points.
Step 1: Calculate the Slope
The slope of a line passing through two points (x₁, y₁) and (x₂, y₂) is given by the formula:
m = (y₂ - y₁) / (x₂ - x₁)
In our case, the given points are (-5, -1) and (1, -5). Let's designate (-5, -1) as (x₁, y₁) and (1, -5) as (x₂, y₂). Substituting these values into the slope formula, we get:
m = (-5 - (-1)) / (1 - (-5))
= (-5 + 1) / (1 + 5)
= -4 / 6
= -2/3
Therefore, the slope of the line passing through the points (-5, -1) and (1, -5) is -2/3. This negative slope indicates that the line slopes downwards from left to right.
Step 2: Use the Point-Slope Form
Now that we have the slope, 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)
We can use either of the given points as (x₁, y₁). Let's use the point (-5, -1). Substituting the slope (-2/3) and the coordinates of the point (-5, -1) into the point-slope form, we get:
y - (-1) = (-2/3)(x - (-5))
y + 1 = (-2/3)(x + 5)
This is the equation of the line in point-slope form. However, it's often desirable to express the equation in slope-intercept form (y = mx + b) or standard form (Ax + By = C). Let's convert the equation to slope-intercept form.
Step 3: Convert to Slope-Intercept Form
To convert the equation from point-slope form to slope-intercept form, we need to isolate 'y' on one side of the equation. Starting with the equation we obtained in the previous step:
y + 1 = (-2/3)(x + 5)
First, distribute the -2/3 on the right side:
y + 1 = (-2/3)x - (2/3)(5)
y + 1 = (-2/3)x - 10/3
Next, subtract 1 from both sides to isolate 'y':
y = (-2/3)x - 10/3 - 1
To combine the constant terms, we need a common denominator. Since 1 can be written as 3/3, we have:
y = (-2/3)x - 10/3 - 3/3
y = (-2/3)x - 13/3
Therefore, the equation of the line in slope-intercept form is:
y = (-2/3)x - 13/3
This equation tells us that the line has a slope of -2/3 and a y-intercept of -13/3.
Step 4: Convert to Standard Form (Optional)
While the slope-intercept form is widely used, the standard form of a linear equation (Ax + By = C) is also common. To convert the equation to standard form, we need to eliminate fractions and rearrange the terms so that 'x' and 'y' are on the same side of the equation and the coefficients are integers. Starting with the slope-intercept form:
y = (-2/3)x - 13/3
Multiply both sides of the equation by 3 to eliminate the fractions:
3y = 3((-2/3)x - 13/3)
3y = -2x - 13
Add 2x to both sides to move the 'x' term to the left side:
2x + 3y = -13
Therefore, the equation of the line in standard form is:
2x + 3y = -13
This equation satisfies the standard form criteria, where A = 2, B = 3, and C = -13.
Summary of Results
We have successfully determined the equation of the line passing through the points (-5, -1) and (1, -5) in three different forms:
- Slope-intercept form:
y = (-2/3)x - 13/3 - Point-slope form:
y + 1 = (-2/3)(x + 5) - Standard form:
2x + 3y = -13
Each of these equations represents the same line, but they offer different perspectives and are useful in various contexts. The slope-intercept form highlights the slope and y-intercept, the point-slope form is convenient when a point and slope are known, and the standard form is often used for general representation and algebraic manipulations.
Conclusion
Finding the equation of a line passing through two points is a fundamental concept in coordinate geometry. By calculating the slope and utilizing the point-slope form or slope-intercept form, we can derive the equation of the line. This article has provided a comprehensive step-by-step guide, illustrating the process with a specific example. Understanding these concepts is crucial for further exploration of linear equations and their applications in mathematics and various fields.
Whether you're a student learning the basics or a professional applying these concepts in real-world scenarios, mastering the techniques for finding the equation of a line is an invaluable skill. The ability to represent linear relationships mathematically allows for analysis, prediction, and problem-solving in a wide range of contexts. Remember to practice and apply these concepts to solidify your understanding and build confidence in your mathematical abilities.
