Equation Of A Line Passing Through (-8,-9) With Slope -3/4 In Slope-Intercept Form
Introduction to Slope-Intercept Form
In the fascinating world of mathematics, linear equations reign supreme as fundamental tools for describing relationships between variables. Among the various forms of linear equations, the slope-intercept form stands out for its simplicity and interpretability. This form, expressed as y = mx + b, elegantly reveals two crucial characteristics of a line: its slope (m) and its y-intercept (b). The slope, often described as "rise over run," quantifies the steepness and direction of the line, while the y-intercept pinpoints the point where the line intersects the vertical axis. Mastering the slope-intercept form is essential for anyone seeking to understand and manipulate linear relationships, and it lays the groundwork for more advanced mathematical concepts.
In this comprehensive exploration, we embark on a journey to unravel the process of constructing the equation of a line in slope-intercept form. Our specific challenge is to determine the equation of a line that gracefully passes through the point (-8, -9) and possesses a slope of -3/4. This seemingly simple problem encapsulates the essence of linear equations and provides a valuable opportunity to hone our analytical skills. By systematically applying the principles of slope-intercept form and leveraging the given information, we will arrive at the precise equation that defines this unique line. This endeavor not only reinforces our understanding of linear equations but also equips us with the tools to tackle similar challenges with confidence and precision.
The journey begins with a clear understanding of the problem at hand. We are given a point, (-8, -9), which the line must traverse, and the slope, -3/4, which dictates its inclination. Our mission is to weave these two pieces of information into the fabric of the slope-intercept form, y = mx + b. The slope, m, is already known to us, but the y-intercept, b, remains an elusive unknown. To unveil this mystery, we will employ a strategic maneuver: substituting the coordinates of the given point and the slope into the slope-intercept equation. This substitution will transform the equation into a solvable puzzle, where b emerges as the sole unknown, beckoning us to find its value. Once we have successfully determined b, we will have all the necessary ingredients to assemble the complete equation of the line in slope-intercept form.
Point-Slope Form: A Stepping Stone
Before we directly tackle the slope-intercept form, it is beneficial to introduce another valuable tool in our mathematical arsenal: the point-slope form. This form, expressed as y - y₁ = m(x - x₁), provides an alternative pathway to defining a line, particularly when we know a point on the line (x₁, y₁) and its slope (m). The point-slope form elegantly captures the relationship between the slope, a known point, and any other point (x, y) on the line. It serves as a powerful stepping stone towards the slope-intercept form, as we can easily transform it through algebraic manipulation.
In our quest to find the equation of the line, we can leverage the point-slope form as an intermediate stage. By substituting the given point (-8, -9) and the slope -3/4 into the point-slope form, we obtain a preliminary equation that embodies the line's characteristics. This equation, while not yet in slope-intercept form, holds all the essential information needed to reach our final destination. The next step involves carefully unraveling the point-slope equation through algebraic simplification. We will distribute the slope, combine like terms, and isolate y on one side of the equation. This meticulous process will gradually transform the equation into the familiar slope-intercept form, revealing the elusive y-intercept and completing our mission.
The point-slope form is not merely a stepping stone; it is a valuable tool in its own right. It provides a flexible and intuitive way to represent linear equations, especially when dealing with specific points and slopes. Understanding and mastering the point-slope form enhances our mathematical versatility and allows us to approach linear equation problems from different angles. In our case, it serves as a bridge between the given information and the desired slope-intercept form, showcasing the interconnectedness of mathematical concepts and the power of strategic problem-solving. By embracing the point-slope form, we gain a deeper appreciation for the elegance and efficiency of linear equations.
Applying the Point-Slope Form
Now, let's put the point-slope form into action. We are given the point (-8, -9), which we will designate as (x₁, y₁), and the slope m = -3/4. Plugging these values into the point-slope form, y - y₁ = m(x - x₁), we get:
y - (-9) = (-3/4)(x - (-8))
This equation, while technically correct, appears somewhat cluttered with negative signs. To simplify it, we can rewrite the subtractions of negative numbers as additions:
y + 9 = (-3/4)(x + 8)
This equation is a more streamlined representation of the line in point-slope form. It encapsulates the essential information about the line's slope and its passage through the point (-8, -9). However, our ultimate goal is to express the equation in slope-intercept form, y = mx + b. To achieve this, we must embark on a journey of algebraic manipulation, carefully transforming the equation while preserving its fundamental meaning. The next step involves distributing the slope, -3/4, across the parentheses on the right side of the equation.
The distribution step is crucial for unraveling the point-slope equation and bringing it closer to the desired slope-intercept form. By multiplying -3/4 by both x and 8, we begin to separate the terms and pave the way for isolating y. This process requires careful attention to detail, ensuring that we correctly apply the distributive property and maintain the integrity of the equation. As we distribute the slope, we are essentially expanding the equation, revealing its underlying structure and preparing it for further simplification. This step-by-step approach is a hallmark of mathematical problem-solving, where complex tasks are broken down into manageable steps, each building upon the previous one. By mastering the distribution step, we gain a valuable skill that extends beyond linear equations, empowering us to tackle a wide range of algebraic challenges.
Transforming to Slope-Intercept Form
Having successfully applied the point-slope form and simplified the equation to y + 9 = (-3/4)(x + 8), our next crucial step is to transform this equation into the coveted slope-intercept form, y = mx + b. This transformation involves a series of algebraic manipulations, each designed to isolate y on one side of the equation and reveal the y-intercept, b. The first step in this process is to distribute the slope, -3/4, across the parentheses on the right side of the equation:
y + 9 = (-3/4)x + (-3/4)(8)
Now, let's simplify the multiplication on the right side:
y + 9 = (-3/4)x - 6
We are getting closer to the slope-intercept form! The next, and final, step is to isolate y by subtracting 9 from both sides of the equation:
y + 9 - 9 = (-3/4)x - 6 - 9
This simplifies to:
y = (-3/4)x - 15
Eureka! We have successfully transformed the equation into slope-intercept form. The equation y = (-3/4)x - 15 represents the line that passes through the point (-8, -9) and has a slope of -3/4. The slope, m, is clearly -3/4, and the y-intercept, b, is -15. This means the line intersects the y-axis at the point (0, -15). We have achieved our goal, demonstrating the power of algebraic manipulation and the elegance of the slope-intercept form.
The journey from the point-slope form to the slope-intercept form highlights the interconnectedness of mathematical concepts. By strategically applying algebraic principles, we can seamlessly transition between different representations of the same linear relationship. This flexibility is a key characteristic of mathematical thinking, allowing us to choose the most appropriate form for a given problem or situation. The ability to transform equations not only deepens our understanding of the underlying concepts but also enhances our problem-solving skills. In this case, we have successfully navigated the algebraic landscape, transforming a point-slope equation into a slope-intercept equation, and in doing so, we have gained a valuable insight into the beauty and power of linear equations.
The Slope-Intercept Equation
After our methodical journey through the realms of point-slope form and algebraic manipulation, we have triumphantly arrived at our destination: the slope-intercept equation of the line. The equation, y = (-3/4)x - 15, stands as a testament to our problem-solving prowess and a clear representation of the line that gracefully passes through the point (-8, -9) while maintaining a slope of -3/4.
This equation, in its elegant simplicity, encapsulates all the essential characteristics of the line. The coefficient of x, -3/4, proudly proclaims the line's slope, indicating its steepness and direction. For every 4 units we move horizontally along the line, we descend 3 units vertically. The constant term, -15, reveals the line's y-intercept, the point where it intersects the vertical axis. This intersection occurs at the coordinates (0, -15), a crucial landmark in the line's trajectory.
The slope-intercept form is not merely a symbolic representation; it is a powerful tool for understanding and visualizing linear relationships. By examining the equation, we can immediately grasp the line's inclination and its position on the coordinate plane. We can effortlessly sketch the line's graph, tracing its path through the point (-8, -9) and extending it infinitely in both directions. The slope-intercept form empowers us to make predictions about the line's behavior, determining the y-value for any given x-value and vice versa. This predictive capability is invaluable in various applications, from modeling real-world phenomena to solving complex mathematical problems.
The equation y = (-3/4)x - 15 is more than just an answer; it is a gateway to a deeper understanding of linear equations. It serves as a concrete example of how slope and y-intercept intertwine to define a line's unique identity. By mastering the process of deriving this equation, we have not only solved a specific problem but also honed our mathematical skills and gained a valuable tool for exploring the world of linear relationships. This equation stands as a symbol of our accomplishment, a testament to the power of mathematical reasoning and the beauty of the slope-intercept form.
Conclusion
In conclusion, we have successfully navigated the mathematical landscape to determine the equation of a line passing through the point (-8, -9) with a slope of -3/4. Our journey began with an exploration of the slope-intercept form, y = mx + b, and the point-slope form, y - y₁ = m(x - x₁). We strategically employed the point-slope form as a stepping stone, substituting the given point and slope to obtain a preliminary equation. Through careful algebraic manipulation, we transformed this equation into the desired slope-intercept form, y = (-3/4)x - 15.
This equation encapsulates the essence of the line, revealing its slope (-3/4) and y-intercept (-15). It serves as a powerful tool for understanding and visualizing the line's behavior, allowing us to predict its trajectory and solve related problems. The process we undertook highlights the interconnectedness of mathematical concepts and the importance of algebraic manipulation in transforming equations into desired forms.
More broadly, this exploration has reinforced the importance of a systematic approach to problem-solving. By breaking down the challenge into smaller, manageable steps, we were able to navigate the complexities of linear equations with confidence and precision. We learned to leverage the strengths of different equation forms, strategically choosing the point-slope form as a bridge to the slope-intercept form. This adaptability is a hallmark of mathematical proficiency, empowering us to tackle a wide range of challenges with creativity and insight.
The journey to find the equation y = (-3/4)x - 15 has been more than just a mathematical exercise; it has been an opportunity to deepen our understanding of linear relationships and enhance our problem-solving skills. The knowledge and skills we have gained will serve us well in future mathematical endeavors, empowering us to explore the world with a more analytical and insightful perspective. The equation stands as a symbol of our accomplishment, a testament to the power of mathematical reasoning and the beauty of linear equations.
