Point-Slope Form Of A Line Equation Explained
We are given that the slope-intercept form of the equation of a line is y = 5x - 3, and this line passes through the point (-2, -13). Our goal is to find the point-slope form of the equation for this same line. The point-slope form is a crucial concept in coordinate geometry, offering a straightforward method to express the equation of a line given a point on the line and its slope. This form is particularly useful when dealing with problems where the slope and a point are readily available, but the y-intercept is not immediately known. The point-slope form not only simplifies the process of writing linear equations but also provides a valuable framework for understanding the geometric properties of lines, such as their steepness and direction. This understanding is fundamental in various mathematical applications, ranging from basic algebra to more advanced calculus and linear algebra. Furthermore, mastering the point-slope form enhances problem-solving skills in real-world scenarios, such as determining the equation of a line representing a linear relationship between two variables in physics, economics, or engineering. This versatile form allows for a deeper comprehension of linear equations and their applications, making it an indispensable tool in mathematical analysis and beyond.
Understanding the Slope-Intercept Form
The slope-intercept form of a linear equation is given by y = mx + b, where m represents the slope of the line and b represents the y-intercept. In our case, the equation is y = 5x - 3. By comparing this to the general form, we can see that the slope (m) is 5 and the y-intercept (b) is -3. The slope, denoted by m, is a critical parameter that defines the steepness and direction of the line. A positive slope indicates that the line rises as you move from left to right, while a negative slope indicates that the line falls. The magnitude of the slope determines how steep the line is; a larger magnitude signifies a steeper line. The y-intercept, denoted by b, is the point where the line intersects the y-axis. This point is crucial for understanding the vertical position of the line on the coordinate plane. Together, the slope and y-intercept provide a comprehensive description of the line's behavior and position. Understanding the slope-intercept form is fundamental for graphing linear equations, identifying key features of the line, and solving problems related to linear relationships. This form is widely used in various mathematical contexts, from basic algebra to calculus, and provides a clear and concise way to express the equation of a line.
The Point-Slope Form
The point-slope form of a linear equation is given by y - y₁ = m(x - x₁), where (x₁, y₁) is a point on the line and m is the slope. This form is particularly useful when we know a point on the line and the slope, but we want to express the equation in a way that directly uses this information. In our problem, we are given the point (-2, -13) and we know the slope is 5 (from the slope-intercept form). The point-slope form, expressed as y - y₁ = m(x - x₁), provides a direct and intuitive way to define a line based on its slope and a specific point it passes through. This form is derived from the definition of slope, which is the change in y divided by the change in x. By rearranging this definition, we can express the equation of a line using the coordinates of a known point (x₁, y₁) and the slope m. This form is especially valuable in situations where the y-intercept is not readily available or when it's more convenient to work with a known point on the line. The point-slope form is not only a practical tool for writing linear equations but also a fundamental concept in understanding the geometry of lines. It allows us to visualize and analyze lines based on their steepness and a reference point, making it an essential tool in various mathematical applications, from basic graphing to advanced calculus and linear algebra.
Applying the Point-Slope Form to Our Problem
To find the point-slope form of the equation for the given line, we substitute the given point (-2, -13) and the slope m = 5 into the point-slope form equation. This means that x₁ = -2, y₁ = -13, and m = 5. Plugging these values into the formula y - y₁ = m(x - x₁), we get:
y - (-13) = 5(x - (-2))
Simplifying this equation, we have:
y + 13 = 5(x + 2)
This is the point-slope form of the equation for the given line. The process of substituting the known values into the point-slope formula is a straightforward application of the formula itself, but it's crucial to pay close attention to the signs of the coordinates. The negative signs in the given point can easily lead to errors if not handled carefully. The resulting equation, y + 13 = 5(x + 2), clearly demonstrates how the point-slope form captures the essential information about the line – its slope and a specific point it passes through. This form allows for easy visualization of the line's behavior and provides a foundation for further analysis. Furthermore, it can be readily converted into other forms of linear equations, such as the slope-intercept form or the standard form, depending on the specific requirements of the problem. Mastering this substitution process is fundamental for effectively utilizing the point-slope form in various mathematical contexts.
Identifying the Correct Option
Now, we compare our result, y + 13 = 5(x + 2), with the given options:
A. y - 13 = 5(x - 2)
B. y + 13 = 5(x + 2)
C. y - 2 = 5(x - 13)
D. y + 2 = 5(x + 13)
By direct comparison, we can see that option B matches our result. The process of comparing the derived equation with the provided options is a critical step in problem-solving, ensuring that the final answer aligns with the given choices. This step not only confirms the correctness of the calculations but also reinforces the understanding of the equation's structure and components. In this case, the correct option is B, y + 13 = 5(x + 2), which matches the point-slope form derived from the given slope and point. The other options, while similar in form, have different signs or values that do not correctly represent the given information. This comparison highlights the importance of careful attention to detail and the ability to accurately identify the matching equation. Furthermore, this step reinforces the conceptual understanding of the point-slope form and its relationship to the slope and the coordinates of a point on the line.
Conclusion
Therefore, the point-slope form of the equation for the line that passes through the point (-2, -13) and has a slope of 5 is y + 13 = 5(x + 2), which corresponds to option B. This problem demonstrates the application of the point-slope form in finding the equation of a line. The ability to transition between different forms of linear equations, such as the slope-intercept form and the point-slope form, is a valuable skill in mathematics. Each form provides a unique perspective on the line and its properties, allowing for flexible problem-solving approaches. The point-slope form, in particular, is highly useful when a point and the slope are known, providing a direct pathway to expressing the equation of the line. Mastering this form not only enhances algebraic skills but also fosters a deeper understanding of the geometry of lines and their representation in the coordinate plane. The process of solving this problem reinforces the importance of carefully applying the formulas, paying attention to signs, and accurately comparing results with the given options. This approach is fundamental for success in various mathematical contexts and beyond.
Final Answer: The final answer is (B)
