Finding Equations Of Parallel Lines A Detailed Explanation
Find the equations that represent the line parallel to 3x - 4y = 7 and passes through the point (-4, -2). Select two options.
In the realm of coordinate geometry, understanding the relationships between lines is fundamental. Parallel lines, in particular, hold a special place, characterized by their unwavering equidistance and shared slope. This article delves into the intricacies of identifying equations that represent lines parallel to a given line and passing through a specific point. We'll dissect the concepts of slope, point-slope form, and slope-intercept form, equipping you with the tools to confidently navigate these geometric challenges. We will specifically address the question of how to find the equations that represent the line parallel to the line $3x - 4y = 7$ and passing through the point $(-4, -2)$. This example will serve as a practical application of the principles discussed, solidifying your understanding and problem-solving skills.
Understanding Parallel Lines and Their Equations
At the heart of parallelism lies the concept of slope. Parallel lines possess the same slope, a crucial property that dictates their unwavering trajectory. The slope, often denoted by 'm', quantifies the steepness and direction of a line. It represents the change in the vertical direction (rise) divided by the change in the horizontal direction (run). For two lines to be parallel, their slopes must be identical. This is the foundational principle upon which we build our understanding of parallel lines and their equations.
To effectively work with linear equations, familiarity with different forms is essential. Two prominent forms are the slope-intercept form and the point-slope form. The slope-intercept form, expressed as $y = mx + b$, explicitly reveals the slope (m) and the y-intercept (b), where the line crosses the vertical axis. This form is particularly useful for visualizing and comparing lines.
The point-slope form, given by $y - y_1 = m(x - x_1)$, provides a powerful alternative when a point on the line $(x_1, y_1)$ and the slope (m) are known. This form allows us to directly construct the equation of a line without explicitly calculating the y-intercept. Both forms are instrumental in determining and manipulating equations of parallel lines.
Converting between these forms is a valuable skill. For instance, we can transform the point-slope form into the slope-intercept form by simply distributing and isolating 'y'. This flexibility empowers us to choose the most convenient form for a given problem, streamlining our calculations and enhancing our understanding of the relationships between lines.
Finding the Slope of the Given Line
Before we can embark on our quest to find parallel lines, we must first decipher the slope of the given line: $3x - 4y = 7$. To achieve this, we'll employ a strategic maneuver: transforming the equation into the slope-intercept form ($y = mx + b$). This form, as we've discussed, readily reveals the slope as the coefficient of 'x'.
Let's embark on this transformation step-by-step. Our initial equation is $3x - 4y = 7$. The first move involves isolating the 'y' term. We subtract $3x$ from both sides, yielding $-4y = -3x + 7$. Now, to completely isolate 'y', we divide both sides by $-4$. This crucial step gives us $y = \frac{3}{4}x - \frac{7}{4}$.
Behold! The equation is now in slope-intercept form. We can clearly identify the slope, 'm', as $\frac{3}{4}$. This value, $\frac{3}{4}$, is the key to unlocking the equations of all lines parallel to the given line. Remember, parallel lines share the same slope, so any line with a slope of $ rac{3}{4}$ will be parallel to $3x - 4y = 7$. This is the cornerstone upon which we will build our solution.
Using the Point-Slope Form to Find the Equation
Now that we've unearthed the slope of the given line ($\frac{3}{4}$), our next objective is to construct the equation of a line parallel to it that gracefully passes through the point $(-4, -2)$. For this task, the point-slope form, $y - y_1 = m(x - x_1)$, emerges as our champion. This form, as we recall, elegantly utilizes a point on the line $(x_1, y_1)$ and the slope (m) to define the line's equation.
Let's plug in the values we know. Our point is $(-4, -2)$, so $x_1 = -4$ and $y_1 = -2$. The slope, as we've established, is $\frac3}{4}$. Substituting these values into the point-slope form, we get{4}(x - (-4))$.
Simplifying this equation is our next step. The double negatives become positives, transforming the equation into $y + 2 = \frac3}{4}(x + 4)$. This equation, while perfectly valid, can be further refined. To obtain the slope-intercept form, we distribute the $ rac{3}{4}$ on the right side{4}x + 3$. Finally, we subtract 2 from both sides to isolate 'y', resulting in $y = \frac{3}{4}x + 1$.
Thus, we have successfully derived the equation of a line parallel to $3x - 4y = 7$ and passing through $(-4, -2)$ in slope-intercept form. This equation, $y = \frac{3}{4}x + 1$, is a key piece of our puzzle. However, it's not the only possible form. We can also express this line in standard form, which we'll explore in the next section.
Converting to Standard Form
While the slope-intercept form ($y = mx + b$) and point-slope form ($y - y_1 = m(x - x_1)$) are invaluable tools, the standard form of a linear equation, $Ax + By = C$, offers a different perspective. In this form, A, B, and C are integers, and A is typically non-negative. Converting our equation into standard form allows us to compare it with other equations in a consistent format.
We begin with our equation in slope-intercept form: $y = \frac{3}{4}x + 1$. The first hurdle is the fraction. To eliminate it, we multiply both sides of the equation by 4, the denominator of the fraction. This yields $4y = 3x + 4$. Now, to achieve the standard form, we need to arrange the terms such that 'x' and 'y' are on the same side and the constant term is on the other side. We subtract $3x$ from both sides, resulting in $-3x + 4y = 4$.
However, standard form convention dictates that the coefficient of 'x' should be non-negative. To rectify this, we multiply the entire equation by $-1$, flipping the signs of all terms. This crucial step gives us $3x - 4y = -4$. We have now successfully transformed our equation into standard form. This form, $3x - 4y = -4$, provides an alternative representation of the line parallel to $3x - 4y = 7$ and passing through $(-4, -2)$.
This standard form equation is a valuable companion to our slope-intercept form equation. It allows us to readily compare the line with other equations expressed in standard form, providing a different lens through which to analyze its properties and relationships with other lines.
Identifying the Correct Options
Now, armed with the knowledge of slope-intercept form ($y = \frac{3}{4}x + 1$) and standard form ($3x - 4y = -4$), we can confidently tackle the original question: Which equations represent the line that is parallel to $3x - 4y = 7$ and passes through the point $(-4, -2)$? We are presented with a set of options, and our mission is to identify the two that match our derived equations.
Let's revisit the options:
By direct comparison, we can immediately recognize that $3x - 4y = -4$ is one of the correct options. This equation precisely matches the standard form equation we derived earlier. Now, let's scrutinize the other options.
The first option, $y = -\frac{3}{4}x + 1$, shares the same y-intercept as our slope-intercept form equation but possesses a different slope. Its slope is $-\frac{3}{4}$, which is the negative reciprocal of our slope $ rac{3}{4}$. This indicates that this line is perpendicular, not parallel, to the given line. Therefore, this option is incorrect.
The third option, $4x - 3y = -10$, requires a bit more investigation. To determine if it represents the same line, we can either convert it to slope-intercept form or check if the point $(-4, -2)$ satisfies the equation. Let's substitute the point's coordinates into the equation: $4(-4) - 3(-2) = -16 + 6 = -10$. The equation holds true for the point. However, to confirm that it represents the same line, we need to check if it has the same slope. Converting to slope-intercept form, we get: $-3y = -4x - 10$ or $y = \frac{4}{3}x + \frac{10}{3}$. The slope, $ rac{4}{3}$, is different from our target slope of $ rac{3}{4}$. Hence, this option is also incorrect.
Therefore, the two equations that represent the line parallel to $3x - 4y = 7$ and passing through the point $(-4, -2)$ are: $y = \frac{3}{4}x + 1$ and $3x - 4y = -4$.
Conclusion
In this exploration of parallel lines, we've traversed the landscape of coordinate geometry, unraveling the significance of slope, point-slope form, and standard form. We've successfully navigated the challenge of finding equations representing lines parallel to a given line and passing through a specific point. The key takeaways are the unwavering relationship between parallel lines and their shared slope, and the power of different equation forms in solving geometric problems. By mastering these concepts, you've equipped yourself with the tools to confidently tackle a wide array of linear equation challenges.
This problem serves as a microcosm of the broader world of coordinate geometry. The principles we've applied here extend to other geometric figures and relationships. The ability to manipulate equations, understand their forms, and connect them to geometric concepts is a cornerstone of mathematical proficiency. As you continue your journey in mathematics, remember the lessons learned here, and they will serve you well in navigating more complex and fascinating mathematical landscapes.