What Is The Equation Of The Line Parallel To The Line Y = (1/5)x + 4 And Passing Through The Point (-2, 2)?
In mathematics, determining the equation of a line that satisfies specific conditions is a fundamental concept in coordinate geometry. This article delves into the process of finding the equation of a line that is parallel to a given line and passes through a specified point. Understanding this concept is crucial for various applications in mathematics, physics, and engineering. We will explore the underlying principles, step-by-step methods, and provide a detailed example to illustrate the process.
Understanding Parallel Lines
To effectively find the equation of a parallel line, it is essential to grasp the concept of parallelism in the context of linear equations. Parallel lines, by definition, are lines that never intersect. In a two-dimensional coordinate plane, this means that parallel lines have the same slope but different y-intercepts. The slope of a line, often denoted by 'm', represents the steepness and direction of the line. It is defined as the ratio of the change in the vertical coordinate (y) to the change in the horizontal coordinate (x), which can be expressed as:
m = (change in y) / (change in x)
In the slope-intercept form of a linear equation, y = mx + b, where 'm' is the slope and 'b' is the y-intercept (the point where the line crosses the y-axis), the slope 'm' is the coefficient of 'x'. The y-intercept, 'b', determines the vertical position of the line on the coordinate plane. Parallel lines share the same slope 'm', ensuring they maintain the same steepness and direction. However, their y-intercepts 'b' differ, preventing them from overlapping or intersecting.
Consider two lines: line 1 with the equation y = m1x + b1 and line 2 with the equation y = m2x + b2. These lines are parallel if and only if m1 = m2 and b1 ≠ b2. This condition ensures that the lines have the same inclination but are vertically shifted relative to each other. Visualizing parallel lines on a graph helps to solidify this concept. Imagine two straight paths running side by side, never converging or diverging. They maintain a constant distance from each other, mirroring the consistent slope and differing y-intercepts of parallel lines in a coordinate plane. This fundamental understanding of parallel lines and their slopes is the foundation for determining the equation of a line parallel to a given line and passing through a specific point.
Determining the Slope of the Parallel Line
The initial step in finding the equation of a line parallel to a given line is to identify the slope of the given line. This is crucial because, as previously discussed, parallel lines share the same slope. The given line is typically presented in one of several forms, each providing a pathway to determine the slope. One common form is the slope-intercept form: y = mx + b, where 'm' represents the slope and 'b' represents the y-intercept. When the equation is in this form, the slope is simply the coefficient of the 'x' term. For instance, if the given line is y = 3x + 2, the slope is directly identified as 3.
Another form in which a line equation may be presented is the standard form: Ax + By = C, where A, B, and C are constants. To determine the slope from this form, the equation needs to be rearranged into the slope-intercept form. This involves isolating 'y' on one side of the equation. By subtracting Ax from both sides and then dividing by B, the equation transforms into y = (-A/B)x + (C/B). From this rearranged form, the slope 'm' is identified as -A/B. For example, if the equation is 2x + 3y = 6, rearranging it gives y = (-2/3)x + 2, revealing a slope of -2/3.
In some instances, the equation may be presented in the point-slope form: y - y1 = m(x - x1), where 'm' is the slope and (x1, y1) is a point on the line. In this form, the slope 'm' is explicitly given. For example, in the equation y - 4 = 2(x - 1), the slope is immediately recognized as 2. Once the given line's equation is in any of these forms, extracting the slope becomes a straightforward task. The slope obtained is the same slope that the parallel line will have. This slope, along with a given point through which the parallel line passes, will be used in the subsequent steps to formulate the equation of the parallel line.
Using the Point-Slope Form
Having determined the slope of the parallel line, the next step involves using the point-slope form of a linear equation to construct the equation of the desired line. The point-slope form is a versatile and efficient way to express the equation of a line when a point on the line and the slope are known. It is given by:
y - y1 = m(x - x1)
where 'm' represents the slope of the line, and (x1, y1) are the coordinates of a known point on the line. In the context of finding a parallel line, 'm' is the slope obtained from the given line (as parallel lines have the same slope), and (x1, y1) is the given point through which the parallel line must pass. To apply the point-slope form, substitute the known values of 'm', x1, and y1 into the equation. This substitution results in an equation that represents the line with the specified slope and passing through the given point. For instance, if the slope 'm' is 2 and the line passes through the point (1, 3), the substitution yields:
y - 3 = 2(x - 1)
This equation is a valid representation of the line, but it is often desirable to express the equation in a more standard form, such as the slope-intercept form (y = mx + b) or the standard form (Ax + By = C). To convert the equation from point-slope form to slope-intercept form, distribute the slope 'm' on the right side of the equation and then isolate 'y' on the left side. Continuing with the previous example:
y - 3 = 2x - 2
Adding 3 to both sides gives:
y = 2x + 1
This is the equation of the line in slope-intercept form, clearly showing the slope and y-intercept. Alternatively, to convert to standard form, rearrange the terms such that 'x' and 'y' are on the same side of the equation and the constant term is on the other side. From the slope-intercept form y = 2x + 1, subtracting 2x from both sides gives:
-2x + y = 1
Multiplying by -1 to make the coefficient of 'x' positive gives:
2x - y = -1
This is the equation of the line in standard form. The point-slope form provides a direct and efficient method for writing the equation of a line when the slope and a point are known, and it can be easily converted to other forms as needed.
Converting to Slope-Intercept Form
While the point-slope form provides a direct method for writing the equation of a line, the slope-intercept form (y = mx + b) is often preferred for its clarity and ease of interpretation. Converting the equation from point-slope form to slope-intercept form involves a few algebraic manipulations. The primary goal is to isolate 'y' on one side of the equation, expressing it in terms of 'x' and a constant. This process typically consists of two main steps: distributing the slope and then isolating 'y'. Starting with the point-slope form:
y - y1 = m(x - x1)
The first step is to distribute the slope 'm' across the terms inside the parentheses on the right side of the equation. This means multiplying 'm' by both 'x' and -x1. The equation then becomes:
y - y1 = mx - mx1
Next, the goal is to isolate 'y' on the left side of the equation. This is achieved by adding y1 to both sides of the equation. The '-y1' on the left side cancels out, leaving 'y' by itself:
y = mx - mx1 + y1
This equation is now in slope-intercept form, y = mx + b, where 'm' is the slope and 'b' is the y-intercept. The y-intercept, 'b', is represented by the expression -mx1 + y1. This form clearly shows the slope of the line and the point where the line intersects the y-axis. For example, consider a line with a slope of 3 that passes through the point (2, 4). Using the point-slope form:
y - 4 = 3(x - 2)
Distributing the slope:
y - 4 = 3x - 6
Isolating 'y'**:
y = 3x - 6 + 4
Simplifying:
y = 3x - 2
This final equation is in slope-intercept form, revealing a slope of 3 and a y-intercept of -2. Converting to slope-intercept form not only provides a clear understanding of the line's characteristics but also facilitates graphing the line and comparing it with other lines. The slope-intercept form is a standard and widely used representation of linear equations, making it a valuable tool in various mathematical and practical applications.
Example: Finding the Equation of a Parallel Line
Let's illustrate the process of finding the equation of a line that is parallel to a given line and passes through a specific point with a concrete example. Suppose we are given the line:
y = (1/5)x + 4
and we want to find the equation of a line that is parallel to this line and passes through the point (-2, 2). Following the steps outlined earlier, we first need to determine the slope of the given line. The equation y = (1/5)x + 4 is in slope-intercept form, y = mx + b, where 'm' is the slope and 'b' is the y-intercept. By comparing the given equation with the slope-intercept form, we can identify the slope as 1/5. Since parallel lines have the same slope, the line we are trying to find will also have a slope of 1/5.
Next, we use the point-slope form of a linear equation to construct the equation of the parallel line. The point-slope form is:
y - y1 = m(x - x1)
where 'm' is the slope and (x1, y1) is the given point. We know the slope m = 1/5 and the point (-2, 2), so we substitute these values into the point-slope form:
y - 2 = (1/5)(x - (-2))
Simplifying the equation gives:
y - 2 = (1/5)(x + 2)
This equation represents the line in point-slope form. However, it is often more convenient to express the equation in slope-intercept form, y = mx + b. To convert the equation to slope-intercept form, we distribute the slope (1/5) on the right side of the equation:
y - 2 = (1/5)x + (1/5)(2)
y - 2 = (1/5)x + 2/5
Then, we isolate 'y' by adding 2 to both sides of the equation:
y = (1/5)x + 2/5 + 2
To add the constants, we need a common denominator. We can rewrite 2 as 10/5:
y = (1/5)x + 2/5 + 10/5
Combining the constants gives:
y = (1/5)x + 12/5
This is the equation of the line in slope-intercept form. The line has a slope of 1/5 and a y-intercept of 12/5. Therefore, the equation of the line that is parallel to y = (1/5)x + 4 and passes through the point (-2, 2) is y = (1/5)x + 12/5. This example demonstrates the step-by-step process of finding the equation of a parallel line, starting from identifying the slope to converting the equation to slope-intercept form.
Conclusion
In summary, finding the equation of a line parallel to a given line and passing through a specific point is a fundamental concept in coordinate geometry. The process involves understanding that parallel lines share the same slope, using the point-slope form to construct the equation, and converting it to slope-intercept form for clarity. By following these steps, one can effectively determine the equation of any parallel line. Mastering this concept is crucial for various mathematical applications and problem-solving scenarios.