Equation Of A Line Parallel To Y-1=4(x+3) Passing Through (4,32)

Jul 10, 2025 by ADMIN 65 views

Finding the Equation of a Parallel Line Passing Through a Given Point

In mathematics, determining the equation of a line that is parallel to a given line and passes through a specific point is a fundamental concept in coordinate geometry. This problem involves understanding the properties of parallel lines, the slope-intercept form of a linear equation, and how to use a point and a slope to define a unique line. This comprehensive guide will walk you through the steps to solve such problems, using the example of finding the equation of a line parallel to ${ y - 1 = 4(x + 3) }$ and passing through the point ${ (4, 32) }$ . Let's delve into the intricacies of linear equations and parallel lines.

Understanding Parallel Lines and Their Slopes

To effectively tackle the problem at hand, it's crucial to grasp the concept of parallel lines and their slopes. Parallel lines, by definition, are lines that lie in the same plane but never intersect. A fundamental property of parallel lines is that they have the same slope. The slope of a line, often denoted by ${ m }$ , represents the steepness and direction of the line. It is the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line.

The equation of a line can be expressed in various forms, but the slope-intercept form is particularly useful for identifying the slope. The slope-intercept form is given by:

${ y = mx + b }$

where:

${ y }$ is the dependent variable (usually plotted on the vertical axis).
${ x }$ is the independent variable (usually plotted on the horizontal axis).
${ m }$ is the slope of the line.
${ b }$ is the y-intercept, which is the point where the line crosses the y-axis.

In the context of parallel lines, if we have two lines, say line 1 and line 2, and they are parallel, then their slopes are equal. Mathematically, this can be expressed as:

${ m_1 = m_2 }$

where ${ m_1 }$ is the slope of line 1 and ${ m_2 }$ is the slope of line 2. Understanding this relationship is crucial for solving problems involving parallel lines.

When dealing with linear equations, recognizing the slope is essential for various applications, including determining the relationship between lines, predicting the behavior of linear functions, and solving geometric problems. For instance, in our case, the given line ${ y - 1 = 4(x + 3) }$ needs to be converted into slope-intercept form to identify its slope. This slope will then be used to find the equation of the parallel line. By converting the equation into slope-intercept form, we can easily see the slope as the coefficient of ${ x }$ . This makes the process of finding the equation of a parallel line more straightforward, as we already have one crucial piece of information: the slope. Moreover, understanding the significance of the y-intercept helps in visualizing the line's position on the coordinate plane and further aids in solving related problems. The ability to interpret and manipulate linear equations is a foundational skill in algebra and geometry, enabling us to solve a wide array of mathematical problems efficiently.

Converting the Given Equation to Slope-Intercept Form

To determine the equation of a line parallel to a given line, the first step is to identify the slope of the given line. The given equation is ${ y - 1 = 4(x + 3) }$ . To find the slope, we need to convert this equation into the slope-intercept form, which is ${ y = mx + b }$ , where ${ m }$ represents the slope and ${ b }$ represents the y-intercept. The slope-intercept form makes it easy to read off the slope directly from the equation.

Let's convert the given equation step by step:

Start with the given equation: ${ y - 1 = 4(x + 3) }$
Distribute the 4 on the right side of the equation: ${ y - 1 = 4x + 12 }$
Add 1 to both sides of the equation to isolate ${ y }$ on the left side: ${ y - 1 + 1 = 4x + 12 + 1 }$
Simplify the equation: ${ y = 4x + 13 }$

Now the equation is in the slope-intercept form ${ y = mx + b }$ . By comparing this with our equation ${ y = 4x + 13 }$ , we can identify the slope ${ m }$ as 4. This means that the given line has a slope of 4. Since parallel lines have the same slope, any line parallel to this line will also have a slope of 4. This understanding is crucial for the next steps in finding the equation of the parallel line. By recognizing the slope, we can use this information along with the given point to determine the unique equation of the parallel line, ensuring it meets the specified conditions.

The ability to convert equations into slope-intercept form is a fundamental skill in algebra. It allows us to quickly identify key features of the line, such as its slope and y-intercept, which are essential for graphing the line and understanding its behavior. Moreover, this skill is not only useful in academic settings but also has practical applications in various fields, including engineering, economics, and computer science. For example, in physics, understanding the slope of a line can help analyze motion graphs, and in economics, it can be used to interpret supply and demand curves. By mastering the technique of converting equations into slope-intercept form, students can enhance their problem-solving abilities and gain a deeper understanding of linear relationships. This foundational knowledge opens doors to more advanced mathematical concepts and real-world applications.

Using the Point-Slope Form to Find the Equation

Now that we know the slope of the parallel line is 4, we can use the point-slope form of a linear equation to find the equation of the line that passes through the point ${ (4, 32) }$ . The point-slope form is a versatile way to represent a line when we have a point on the line and its slope. It is given by:

${ y - y_1 = m(x - x_1) }$

where:

${ (x_1, y_1) }$ is a known point on the line.
${ m }$ is the slope of the line.

In our case, we have the point ${ (4, 32) }$ , so ${ x_1 = 4 }$ and ${ y_1 = 32 }$ . We also know that the slope ${ m }$ is 4, as the line is parallel to the line ${ y = 4x + 13 }$ . Now, we can substitute these values into the point-slope form:

${ y - 32 = 4(x - 4) }$

This equation represents the line that is parallel to the given line and passes through the point ${ (4, 32) }$ . However, to match the answer choices provided, we need to convert this equation into the slope-intercept form ${ y = mx + b }$ . Let's simplify the equation:

Distribute the 4 on the right side: ${ y - 32 = 4x - 16 }$
Add 32 to both sides to isolate ${ y }$ : ${ y - 32 + 32 = 4x - 16 + 32 }$
Simplify the equation: ${ y = 4x + 16 }$

Thus, the equation of the line that is parallel to the line ${ y - 1 = 4(x + 3) }$ and passes through the point ${ (4, 32) }$ is ${ y = 4x + 16 }$ . This matches option D in the given choices. Using the point-slope form allows us to efficiently construct the equation of a line when we have a point and the slope, making it a valuable tool in coordinate geometry problems. The ability to manipulate and simplify these equations is crucial for arriving at the correct solution and understanding the relationship between different forms of linear equations. This skill enhances problem-solving capabilities and provides a solid foundation for more advanced mathematical concepts.

Verifying the Solution

To ensure the accuracy of our solution, it's essential to verify that the equation ${ y = 4x + 16 }$ indeed represents a line parallel to the given line ${ y - 1 = 4(x + 3) }$ and passes through the point ${ (4, 32) }$ . Verification not only confirms the correctness of our calculations but also reinforces our understanding of the underlying concepts.

Checking for Parallelism

We've already established that parallel lines have the same slope. The given line ${ y - 1 = 4(x + 3) }$ was converted to slope-intercept form as ${ y = 4x + 13 }$ , which has a slope of 4. Our solution, ${ y = 4x + 16 }$ , also has a slope of 4. Since both lines have the same slope, they are indeed parallel. This confirms the first condition for our solution to be correct.

Checking if the Point Lies on the Line

To verify that the line ${ y = 4x + 16 }$ passes through the point ${ (4, 32) }$ , we need to substitute the coordinates of the point into the equation and see if the equation holds true. The point ${ (4, 32) }$ has ${ x = 4 }$ and ${ y = 32 }$ . Substituting these values into the equation:

${ 32 = 4(4) + 16 }$

Let's simplify the right side of the equation:

${ 32 = 16 + 16 }$

${ 32 = 32 }$

The equation holds true, which means the point ${ (4, 32) }$ lies on the line ${ y = 4x + 16 }$ . This confirms the second condition for our solution to be correct.

By verifying both the parallelism and the point lying on the line, we can confidently conclude that our solution ${ y = 4x + 16 }$ is correct. This process of verification is crucial in mathematics as it ensures that our solution satisfies all the given conditions and constraints. Furthermore, it helps in identifying any potential errors in our calculations or reasoning. Developing a habit of verifying solutions is a valuable skill that enhances problem-solving abilities and promotes accuracy in mathematical work. It reinforces the understanding of the concepts involved and builds confidence in the results obtained.

Conclusion

In summary, finding the equation of a line that is parallel to a given line and passes through a specific point involves several key steps. First, we need to identify the slope of the given line by converting it into slope-intercept form. Then, we use the fact that parallel lines have the same slope to determine the slope of the line we want to find. Next, we use the point-slope form of a linear equation, along with the given point and the slope, to construct the equation of the parallel line. Finally, we simplify the equation into slope-intercept form to match the standard format and verify our solution to ensure accuracy.

In this specific case, we found that the equation of the line parallel to ${ y - 1 = 4(x + 3) }$ and passing through the point ${ (4, 32) }$ is ${ y = 4x + 16 }$ , which corresponds to option D. By understanding the properties of parallel lines and mastering the techniques of converting equations and using the point-slope form, we can effectively solve a wide range of problems in coordinate geometry. This process not only helps in finding the correct answer but also enhances our understanding of linear equations and their applications. The ability to manipulate and interpret linear equations is a fundamental skill in mathematics, with applications extending to various fields, including physics, engineering, economics, and computer science. By practicing and applying these concepts, students can develop a strong foundation in mathematics and improve their problem-solving abilities.