Finding Lines With The Same Slope A Comprehensive Guide
In mathematics, particularly in coordinate geometry, the slope of a line is a crucial concept that defines its steepness and direction. Understanding how to calculate and compare slopes is fundamental for various applications, including determining parallel lines, analyzing linear functions, and solving geometric problems. This article delves into the process of finding a line with the same slope as a given line, using a specific example to illustrate the key steps and concepts involved.
Understanding Slope
Before we dive into the problem, let's recap the definition of slope. The slope, often denoted by m, is a measure of how much a line rises or falls for every unit of horizontal change. Mathematically, it is calculated as the ratio of the change in the vertical coordinate (Δy) to the change in the horizontal coordinate (Δx) between any two points on the line. The formula for the slope (m) given two points (x₁, y₁) and (x₂, y₂) is:
m = (y₂ - y₁) / (x₂ - x₁)
A positive slope indicates an increasing line (from left to right), a negative slope indicates a decreasing line, a slope of zero represents a horizontal line, and an undefined slope signifies a vertical line. The concept of slope is central to understanding linear equations and their graphical representations. It allows us to describe the steepness and direction of a line, which is crucial in many real-world applications, such as determining the pitch of a roof or the grade of a road.
Calculating the Slope of a Line Passing Through Two Points
Our problem begins with identifying the slope of a line passing through two given points: (-4, -1) and (-1, 5). To find the slope, we apply the formula mentioned above. Let's label our points: (x₁, y₁) = (-4, -1) and (x₂, y₂) = (-1, 5). Plugging these values into the slope formula, we get:
m = (5 - (-1)) / (-1 - (-4))
Simplifying the expression:
m = (5 + 1) / (-1 + 4)
m = 6 / 3
m = 2
Therefore, the slope of the line passing through the points (-4, -1) and (-1, 5) is 2. This means that for every one unit we move to the right along the line, we move two units upwards. Understanding this calculation is the foundation for solving the rest of the problem, as we now need to identify which of the given options has the same slope.
Analyzing Option A: y = (1/2)x + 5
The first option provided is the equation y = (1/2)x + 5. This equation is in slope-intercept form, which is a standard way to represent linear equations. The slope-intercept form is given by y = mx + b, where m represents the slope and b represents the y-intercept (the point where the line crosses the y-axis). In this equation, the coefficient of x is the slope. Therefore, by comparing y = (1/2)x + 5 with the slope-intercept form, we can directly identify the slope.
In the equation y = (1/2)x + 5, the coefficient of x is 1/2. This means that the slope of this line is 1/2. To determine if this line has the same slope as the line passing through (-4, -1) and (-1, 5), we compare this slope (1/2) with the slope we calculated earlier (2). Since 1/2 is not equal to 2, the line represented by the equation y = (1/2)x + 5 does not have the same slope as the given line. This simple comparison allows us to quickly eliminate option A from our possible solutions.
Analyzing Option B: 6x - 3y = 5
The second option is the equation 6x - 3y = 5. This equation is in standard form, which is another common way to represent linear equations. The standard form is generally written as Ax + By = C, where A, B, and C are constants. To determine the slope of this line, we need to convert it into slope-intercept form (y = mx + b). This involves isolating y on one side of the equation.
Starting with the equation 6x - 3y = 5, we first subtract 6x from both sides to isolate the term with y:
-3y = -6x + 5
Next, we divide both sides by -3 to solve for y:
y = (-6x + 5) / -3
Distributing the division, we get:
y = (-6x / -3) + (5 / -3)
Simplifying the equation:
y = 2x - 5/3
Now the equation is in slope-intercept form (y = mx + b). By comparing this with the general form, we can see that the slope m is 2. Since the slope of this line (2) is the same as the slope of the line passing through (-4, -1) and (-1, 5), this option is a potential answer. However, we need to examine the remaining options to ensure we find the correct one.
Analyzing Option C: The Line Passing Through (1, 4) and (1, 6)
Option C describes a line passing through the points (1, 4) and (1, 6). To find the slope of this line, we use the slope formula again. Let (x₁, y₁) = (1, 4) and (x₂, y₂) = (1, 6). Plugging these values into the slope formula, we get:
m = (6 - 4) / (1 - 1)
Simplifying the expression:
m = 2 / 0
Here, we encounter a division by zero, which is undefined in mathematics. This means that the slope of the line passing through (1, 4) and (1, 6) is undefined. Lines with undefined slopes are vertical lines. Since the slope of the line passing through (-4, -1) and (-1, 5) is 2, which is a defined value, the line described in option C does not have the same slope. Therefore, we can eliminate option C.
Analyzing Option D: y = 2
Option D presents the equation y = 2. This equation represents a horizontal line. In the coordinate plane, a horizontal line is a line where the y-coordinate is constant for all x-values. To understand the slope of a horizontal line, we can consider any two points on the line. For example, let's take the points (0, 2) and (1, 2), which both lie on the line y = 2. Using the slope formula:
m = (2 - 2) / (1 - 0)
Simplifying the expression:
m = 0 / 1
m = 0
The slope of the line y = 2 is 0. Since the slope of the line passing through (-4, -1) and (-1, 5) is 2, which is not equal to 0, the line represented by the equation y = 2 does not have the same slope. Consequently, we can eliminate option D.
Conclusion
After analyzing all the options, we have determined that only option B, the line represented by the equation 6x - 3y = 5, has the same slope as the line passing through the points (-4, -1) and (-1, 5). By systematically calculating the slopes of each given line and comparing them, we were able to arrive at the correct answer. This exercise highlights the importance of understanding the concept of slope and being able to manipulate linear equations into different forms to identify their properties. Mastering these skills is crucial for success in algebra and beyond.
Key takeaways from this problem include:
- The slope formula: m = (y₂ - y₁) / (x₂ - x₁)
- Slope-intercept form: y = mx + b, where m is the slope
- Standard form: Ax + By = C
- Horizontal lines have a slope of 0.
- Vertical lines have an undefined slope.
By understanding these concepts and practicing similar problems, you can build a strong foundation in coordinate geometry and linear equations. This knowledge is not only valuable for academic success but also for real-world applications where understanding the relationships between lines and their slopes is essential.