Finding The Equation Of A Line Parallel To Y = 2x + 3 Passing Through (a, 6)
In the realm of coordinate geometry, one of the fundamental concepts is understanding and determining the equations of lines. Lines, as we know, are defined by their slope and a point they pass through. When we talk about parallel lines, we introduce an interesting relationship where these lines share the same slope but have different y-intercepts. This article delves into the process of finding the equation of a line that is parallel to another given line and passes through a specific point. We will explore the underlying principles, the steps involved, and provide a detailed example to illustrate the method. Understanding these concepts is crucial not only for mathematics students but also for anyone working with geometric representations in various fields like engineering, computer graphics, and data visualization.
Understanding Parallel Lines
Before we dive into the problem, it's essential to grasp the concept of parallel lines. In simple terms, parallel lines are lines that run in the same direction and never intersect. A key characteristic of parallel lines is that they have the same slope. The slope of a line, often denoted by 'm', represents the steepness or inclination of the line. It is the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line. Mathematically, if we have two points (x1, y1) and (x2, y2) on a line, the slope 'm' is calculated as:
m = (y2 - y1) / (x2 - x1)
The equation of a line can be expressed in several forms, but the most commonly used is the slope-intercept form, which is:
y = mx + c
where 'm' is the slope and 'c' is the y-intercept (the point where the line crosses the y-axis). When two lines are parallel, their slopes (m) are equal. However, their y-intercepts (c) are different, otherwise, they would be the same line. This understanding forms the basis for solving problems involving parallel lines.
Problem Statement
Our main objective in this article is to determine the equation of a line, let's call it line L, that satisfies two specific conditions. First, line L must pass through a given point P, which has coordinates (a, 6). Second, line L must be parallel to another given line, which is defined by the equation y = 2x + 3. This problem combines our understanding of parallel lines and the point-slope form of a line equation. To solve this, we will first identify the slope of the given line. Since parallel lines have the same slope, we can use this slope for our line L. Then, we will use the given point P and the slope to find the equation of line L. This involves substituting the known values into the point-slope form or the slope-intercept form and solving for the unknowns. The process is a blend of algebraic manipulation and geometric understanding, making it a classic example of how mathematics connects different concepts.
Method 1: Using Slope-Intercept Form (y = mx + c)
Our first approach to finding the equation of line L involves using the slope-intercept form, which is a widely recognized and straightforward method. The slope-intercept form of a linear equation is expressed as:
y = mx + c
Where 'm' represents the slope of the line, and 'c' represents the y-intercept (the point where the line crosses the y-axis). This form is particularly useful because it directly gives us the slope and the y-intercept of the line. To apply this method to our problem, we will follow a series of logical steps, starting with identifying the slope of the given parallel line and then using the given point to find the y-intercept of line L. This method is a clear and concise way to determine the equation of a line, and it highlights the significance of the slope and y-intercept in defining a line's position and orientation in the coordinate plane.
Step 1: Determine the Slope of the Given Line
The first critical step in finding the equation of line L is to determine the slope of the line that it is parallel to. We are given the equation of this line as y = 2x + 3. This equation is already in slope-intercept form, which makes our task much easier. By comparing the given equation with the general form y = mx + c, we can directly identify the slope. In this case, the coefficient of x is 2, which means the slope of the given line is 2. Since line L is parallel to this line, it will have the same slope. This is a fundamental property of parallel lines: they have equal slopes. Thus, the slope of line L, which we will denote as mL, is also 2. This step is crucial because the slope is a key component in defining the direction and steepness of a line, and it will be used in subsequent steps to find the full equation of line L.
Step 2: Use the Point-Slope Form to Find the Equation
Now that we have determined the slope of line L, which is 2, we need to find its equation. We know that line L passes through the point P(a, 6). To find the equation, we can use the point-slope form of a linear equation. The point-slope form is particularly useful when we have a point on the line and the slope of the line. It is given by:
y - y1 = m(x - x1)
Where (x1, y1) is a point on the line, and m is the slope. In our case, (x1, y1) is the point P(a, 6), and m is 2. Substituting these values into the point-slope form, we get:
y - 6 = 2(x - a)
This equation represents line L in point-slope form. However, to express the equation in slope-intercept form (y = mx + c), we need to simplify and rearrange the equation. This involves distributing the 2 on the right side and then isolating y on the left side. This step is a critical transition from a form that uses a specific point to a more general form that shows the line's slope and y-intercept. The algebraic manipulation here is a key skill in coordinate geometry.
Step 3: Convert to Slope-Intercept Form (y = mx + c)
After applying the point-slope form, we have the equation y - 6 = 2(x - a). The next step is to convert this equation to slope-intercept form, which is y = mx + c. This form will explicitly show us the slope and the y-intercept of line L. To do this, we need to simplify and rearrange the equation. First, we distribute the 2 on the right side of the equation:
y - 6 = 2x - 2a
Next, we isolate y by adding 6 to both sides of the equation:
y = 2x - 2a + 6
Now, the equation is in the form y = mx + c, where m is the slope and c is the y-intercept. In this case, the slope m is 2 (as we already knew), and the y-intercept c is -2a + 6. This equation, y = 2x - 2a + 6, is the equation of line L in slope-intercept form. It shows how the line's position is influenced by the value of 'a', which is the x-coordinate of the point P. This final step completes our solution using the slope-intercept form method.
Method 2: Using the Point-Slope Form Directly
Our second method to determine the equation of line L involves a more direct application of the point-slope form. This approach is particularly efficient when we already have the slope and a point on the line, which is exactly our situation. The point-slope form, as we discussed earlier, is given by:
y - y1 = m(x - x1)
Where (x1, y1) is a known point on the line, and m is the slope of the line. In this method, we will substitute the known values directly into this form and simplify the equation to obtain the equation of line L. This method is advantageous for its conciseness and its direct use of the given information. It also reinforces the understanding of how the point-slope form encapsulates the relationship between a line's slope, a point on the line, and the line's equation. This approach is a valuable tool in solving linear equation problems and highlights the versatility of the point-slope form.
Step 1: Apply the Point-Slope Form
In this method, we directly apply the point-slope form to find the equation of line L. We know that line L passes through the point P(a, 6) and has a slope of 2 (since it is parallel to the line y = 2x + 3). Using the point-slope form equation:
y - y1 = m(x - x1)
We substitute the known values. Here, (x1, y1) is (a, 6), and m is 2. Plugging these values into the equation, we get:
y - 6 = 2(x - a)
This equation is the point-slope form of line L. It directly represents the line using the given point and slope. The next step is to simplify this equation to a more standard form, such as the slope-intercept form, to make it easier to interpret and use. This direct application of the point-slope form is a powerful technique in coordinate geometry, especially when dealing with problems involving known points and slopes.
Step 2: Simplify the Equation
After applying the point-slope form, we have the equation y - 6 = 2(x - a). The next step is to simplify this equation to obtain a more standard form, such as the slope-intercept form (y = mx + c). To do this, we will distribute the 2 on the right side of the equation and then isolate y on the left side. First, distribute the 2:
y - 6 = 2x - 2a
Next, isolate y by adding 6 to both sides of the equation:
y = 2x - 2a + 6
This is the simplified equation of line L in slope-intercept form. It clearly shows the slope (2) and the y-intercept (-2a + 6). This form is particularly useful because it allows us to easily see how the line's position changes with different values of 'a'. The simplification process here demonstrates a key algebraic skill in manipulating linear equations. This final equation is the solution we were seeking, derived directly from the point-slope form.
Conclusion
In conclusion, we have successfully determined the equation of line L, which passes through the point P(a, 6) and is parallel to the line y = 2x + 3. We explored two different methods to solve this problem, each highlighting different aspects of linear equations and their properties. The first method involved using the slope-intercept form (y = mx + c), where we first identified the slope of the parallel line and then used the given point to find the y-intercept of line L. The second method utilized the point-slope form directly, which is a more concise approach when a point and a slope are known. Both methods led us to the same equation for line L:
y = 2x - 2a + 6
This equation demonstrates that line L has a slope of 2 (parallel to the given line) and a y-intercept of -2a + 6, which varies depending on the value of 'a'. Understanding these methods and the underlying principles of parallel lines and linear equations is crucial for various applications in mathematics, physics, engineering, and computer science. The ability to manipulate and solve linear equations is a fundamental skill that opens doors to more advanced mathematical concepts and real-world problem-solving scenarios.
Parallel lines, Equation of a line, Slope-intercept form, Point-slope form, Coordinate geometry, Linear equations, Slope, Y-intercept, Point, Mathematics
