Finding The Equation Of A Line Slope-Intercept Form Example
In the realm of mathematics, understanding the relationships between points and lines is fundamental. One common way to represent a line is using the slope-intercept form, which provides a clear and concise way to describe its characteristics. This article delves into the process of determining the equation of a line when given a point it passes through and its slope. We'll specifically address the scenario where a line passes through the point (-6, 5) and has a slope of -3/2. By understanding the principles behind the slope-intercept form, you can confidently tackle similar problems and gain a deeper appreciation for the elegance of linear equations. The slope-intercept form is not just a mathematical concept; it's a powerful tool for modeling real-world relationships, from the trajectory of a projectile to the cost of goods based on quantity. Mastering this form opens doors to a wide range of applications and provides a solid foundation for more advanced mathematical concepts. This article aims to break down the process step by step, ensuring that you grasp each concept thoroughly. We'll start by revisiting the basics of the slope-intercept form and then apply it to our specific problem. Remember, mathematics is a journey of understanding, and each step you take builds upon the previous one. So, let's embark on this journey together and unravel the mysteries of linear equations.
Understanding the Slope-Intercept Form
The slope-intercept form of a linear equation is expressed as y = mx + b, where m represents the slope of the line and b represents the y-intercept (the point where the line crosses the y-axis). The slope (m) quantifies the steepness and direction of the line. A positive slope indicates an upward slant from left to right, while a negative slope indicates a downward slant. The magnitude of the slope reflects how quickly the line rises or falls; a larger absolute value means a steeper line. The y-intercept (b) is the value of y when x is 0. It's the point where the line intersects the vertical axis on the coordinate plane. Understanding these two parameters, slope and y-intercept, is crucial for interpreting and manipulating linear equations. The beauty of the slope-intercept form lies in its simplicity and clarity. It directly reveals the two key characteristics of a line, making it easy to visualize and analyze. For instance, if you have the equation y = 2x + 3, you immediately know that the line has a slope of 2 and crosses the y-axis at the point (0, 3). This allows you to quickly sketch the line on a graph or compare it to other lines. Furthermore, the slope-intercept form is invaluable for solving various problems, such as finding the equation of a line given certain information or determining the point of intersection between two lines. Its versatility makes it a cornerstone of linear algebra and a fundamental tool for anyone working with mathematical models. In essence, mastering the slope-intercept form is akin to learning the alphabet of linear equations; it provides the building blocks for more complex concepts and applications.
Applying the Point-Slope Form
When we're given a point and a slope, a particularly useful tool is the point-slope form of a linear equation. This form is expressed as y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope. The point-slope form allows us to directly incorporate the given information into the equation, making it a convenient starting point for finding the slope-intercept form. In our case, we have the point (-6, 5) and a slope of -3/2. Plugging these values into the point-slope form, we get: y - 5 = (-3/2)(x - (-6)). This equation represents the line that passes through the point (-6, 5) and has a slope of -3/2. However, it's not yet in the slope-intercept form. To achieve that, we need to simplify and rearrange the equation. The point-slope form is a powerful intermediary step because it bypasses the need to directly calculate the y-intercept. Instead, it leverages the relationship between the slope, a known point, and any other point on the line. This is particularly useful when the y-intercept is not immediately apparent or easily determined. The point-slope form also highlights the fundamental concept that a line is uniquely defined by its slope and a single point. By fixing the slope and a point, we constrain the line to a specific path on the coordinate plane. This understanding is crucial for grasping the geometric interpretation of linear equations and their applications in various fields. In the next step, we'll focus on transforming the point-slope equation into the slope-intercept form, revealing the y-intercept and providing a complete picture of the line's characteristics.
Converting to Slope-Intercept Form
To convert the equation from point-slope form to slope-intercept form, we need to isolate y on one side of the equation. This involves distributing the slope and then adding the constant term to both sides. Starting with our equation from the previous step, y - 5 = (-3/2)(x + 6), we first distribute the -3/2: y - 5 = (-3/2)x - 9. Next, we add 5 to both sides of the equation to isolate y: y = (-3/2)x - 9 + 5. Simplifying the constant terms, we get: y = (-3/2)x - 4. This is the equation of the line in slope-intercept form. We can now clearly see that the slope of the line is -3/2 and the y-intercept is -4. This means the line crosses the y-axis at the point (0, -4). The process of converting from point-slope to slope-intercept form is a fundamental algebraic manipulation that reinforces the understanding of equation equivalence. By performing these steps, we're not changing the line itself; we're simply expressing its equation in a different, but equally valid, form. This flexibility is crucial for solving various problems and gaining insights into the line's properties. The slope-intercept form provides a clear and concise way to visualize the line's behavior. The negative slope indicates that the line descends from left to right, and the y-intercept tells us where the line crosses the vertical axis. This information can be used to sketch the line on a graph, analyze its relationship with other lines, or even model real-world phenomena. In essence, converting to slope-intercept form is the final step in revealing the complete character of the line, making it readily accessible for further analysis and application.
The Final Equation
Therefore, the equation of the line that passes through the point (-6, 5) and has a slope of -3/2, expressed in slope-intercept form, is y = (-3/2)x - 4. This equation encapsulates all the essential information about the line: its steepness, direction, and point of intersection with the y-axis. The slope of -3/2 tells us that for every 2 units we move to the right along the x-axis, the line descends 3 units along the y-axis. The y-intercept of -4 indicates that the line crosses the y-axis at the point (0, -4). This equation is not just a mathematical formula; it's a concise representation of a geometric object. It allows us to visualize the line's path on the coordinate plane and understand its relationship with other points and lines. Furthermore, this equation can be used to solve various problems, such as finding other points on the line, determining where the line intersects other lines or curves, or even modeling real-world scenarios involving linear relationships. The process of arriving at this final equation highlights the power of algebraic manipulation and the importance of understanding different forms of linear equations. By starting with the point-slope form and converting it to the slope-intercept form, we've demonstrated a systematic approach to solving this type of problem. This approach can be applied to a wide range of similar situations, making it a valuable tool in your mathematical arsenal. In conclusion, the equation y = (-3/2)x - 4 is the culmination of our efforts, providing a complete and insightful representation of the line in question.
Conclusion
In this exploration, we successfully determined the equation of a line passing through the point (-6, 5) with a slope of -3/2. We began by understanding the slope-intercept form (y = mx + b) and its components: slope (m) and y-intercept (b). We then utilized the point-slope form (y - y1 = m(x - x1)) to incorporate the given point and slope into an equation. Finally, we converted the equation to slope-intercept form through algebraic manipulation, arriving at the final answer: y = (-3/2)x - 4. This process underscores the interconnectedness of different forms of linear equations and the power of algebraic manipulation in revealing the underlying properties of lines. The slope-intercept form provides a clear and concise representation of a line, making it easy to visualize, analyze, and apply in various contexts. The ability to find the equation of a line given certain information is a fundamental skill in mathematics, with applications ranging from geometry and calculus to physics and engineering. By mastering this skill, you've equipped yourself with a powerful tool for problem-solving and a deeper understanding of linear relationships. Remember, mathematics is a journey of continuous learning and discovery. Each problem you solve builds upon your existing knowledge and opens doors to new challenges and opportunities. So, continue to explore, experiment, and apply your mathematical skills to the world around you.
