Finding The Equation Of A Parallel Line In Point-Slope Form
In the realm of coordinate geometry, determining the equation of a line that satisfies specific conditions is a fundamental skill. One common scenario involves finding the equation of a line that is parallel to a given line and passes through a particular point. This task leverages the concepts of slope and the point-slope form of a linear equation. In this article, we will delve into the step-by-step process of finding the equation of a line parallel to y = 4x + 3 and passing through the point (-1, 6), expressing the final answer in point-slope form.
Understanding Parallel Lines and Slopes
The cornerstone of this problem lies in the understanding of parallel lines. Parallel lines, by definition, are lines that lie in the same plane and never intersect. A crucial property of parallel lines is that they possess the same slope. The slope of a line, often denoted by 'm', quantifies its steepness or inclination. It represents the change in the vertical direction (y-axis) for every unit change in the horizontal direction (x-axis).
The equation y = 4x + 3 is in slope-intercept form, which is generally expressed as y = mx + c, where 'm' represents the slope and 'c' represents the y-intercept (the point where the line crosses the y-axis). By comparing the given equation with the slope-intercept form, we can readily identify that the slope of the line y = 4x + 3 is 4. Since we are seeking a line parallel to this one, the line we aim to find will also have a slope of 4. This understanding of parallel lines and their slopes is the first key step in solving the problem.
To further elaborate on the concept of slope, consider two points on a line, say (x1, y1) and (x2, y2). The slope 'm' can be calculated using the formula:
m = (y2 - y1) / (x2 - x1)
This formula essentially calculates the 'rise' (change in y) over the 'run' (change in x). A positive slope indicates an upward inclination, while a negative slope indicates a downward inclination. A slope of zero represents a horizontal line, and an undefined slope represents a vertical line.
In the context of parallel lines, the equality of slopes ensures that the lines maintain the same steepness and direction, thus preventing them from ever intersecting. This fundamental property is what allows us to use the slope of the given line (y = 4x + 3) directly in finding the equation of the parallel line.
The Point-Slope Form: A Powerful Tool
The point-slope form of a linear equation is a valuable tool for expressing the equation of a line when we know a point on the line and its slope. The point-slope form is given by:
y - y1 = m(x - x1)
where:
- (x1, y1) represents the coordinates of a known point on the line.
- m represents the slope of the line.
This form is particularly useful because it directly incorporates the slope and a point, making it straightforward to construct the equation of a line given these two pieces of information. Unlike the slope-intercept form (y = mx + c), which requires us to determine the y-intercept, the point-slope form bypasses this step and directly uses a known point.
In our problem, we are given the point (-1, 6) that the parallel line must pass through. This means we have x1 = -1 and y1 = 6. We have also established that the slope of the parallel line is 4 (same as the slope of y = 4x + 3). Now, we have all the necessary components to plug into the point-slope form.
The point-slope form is derived from the concept of slope itself. If we consider any arbitrary point (x, y) on the line and the known point (x1, y1), the slope 'm' can be expressed as:
m = (y - y1) / (x - x1)
Multiplying both sides of this equation by (x - x1) yields the point-slope form:
y - y1 = m(x - x1)
This derivation highlights the fundamental connection between the slope and the point-slope form, reinforcing its validity and utility. The point-slope form provides a direct and efficient way to represent the equation of a line when a point and the slope are known, making it a powerful tool in coordinate geometry.
Applying the Point-Slope Form to Our Problem
Now, let's apply the point-slope form to find the equation of the line parallel to y = 4x + 3 and passing through the point (-1, 6). We have already determined that the slope of the parallel line is 4 (m = 4) and the given point is (-1, 6), which gives us x1 = -1 and y1 = 6. We can now substitute these values directly into the point-slope form:
y - y1 = m(x - x1)
Substituting the values, we get:
y - 6 = 4(x - (-1))
Simplifying the expression inside the parentheses:
y - 6 = 4(x + 1)
This is the equation of the line in point-slope form. It clearly shows the slope (4) and a point on the line (-1, 6). The equation represents all the points (x, y) that lie on the line, and it satisfies the condition of being parallel to the given line y = 4x + 3.
The beauty of the point-slope form is that it directly presents the slope and a point on the line, making it easy to visualize and interpret the equation. If desired, we can further manipulate this equation to convert it into slope-intercept form (y = mx + c) or standard form (Ax + By = C), but the point-slope form itself provides a complete and valid representation of the line.
To further illustrate the point-slope form, let's consider another point on the line. If we were to choose a different point on the line and plug its coordinates into the point-slope form along with the slope, we would obtain an equivalent equation. This demonstrates the flexibility and robustness of the point-slope form in representing a line.
The Final Equation in Point-Slope Form
The final equation of the line parallel to y = 4x + 3 and passing through the point (-1, 6), expressed in point-slope form, is:
y - 6 = 4(x + 1)
This equation encapsulates all the information we need about the line. It tells us that the line has a slope of 4 and passes through the point (-1, 6). The point-slope form is a concise and informative way to represent the equation of a line, and it is particularly useful when we know a point on the line and its slope.
In summary, finding the equation of a line parallel to a given line involves understanding the concept of parallel lines having the same slope and utilizing the point-slope form of a linear equation. By identifying the slope of the given line, using the given point, and substituting these values into the point-slope form, we can readily determine the equation of the parallel line. This process demonstrates the power of coordinate geometry in describing and manipulating geometric objects using algebraic equations.
This equation can be used to find any point on the line. For example, if we want to find the y-coordinate when x = 0, we substitute x = 0 into the equation:
y - 6 = 4(0 + 1) y - 6 = 4 y = 10
So, the point (0, 10) is also on the line. This demonstrates how the point-slope form can be used to generate other points on the line.
In conclusion, by leveraging the principles of parallel lines and the point-slope form, we have successfully determined the equation of the line that meets the specified criteria. This exercise reinforces the interconnectedness of geometric concepts and algebraic representations in the field of mathematics.
