Transforming 2x-3y=-9 Into Slope-Intercept Form A Step-by-Step Guide
In the realm of linear equations, the slope-intercept form stands as a beacon of clarity, offering a direct view into the heart of a line's characteristics: its slope and y-intercept. The equation y = mx + b is not just a formula; it's a powerful tool for visualizing and understanding the behavior of lines on a coordinate plane. Here, m represents the slope, the measure of the line's steepness and direction, and b signifies the y-intercept, the point where the line crosses the vertical axis. Transforming an equation into this form is akin to decoding a secret message, revealing the line's essential properties at a glance.
Consider the equation 2x - 3y = -9, a seemingly unassuming algebraic expression. Yet, within its terms lies the potential to unveil a line's unique identity. Our mission is to embark on a mathematical journey, a series of carefully orchestrated steps, to rewrite this equation in the coveted slope-intercept form. This transformation is not merely a symbolic manipulation; it's a process of revealing the underlying structure of the line, making it easier to graph, compare, and analyze.
The journey begins with isolating the y term, a critical step in our quest. We treat the equation as a delicate balance, where any operation performed on one side must be mirrored on the other. Subtracting 2x from both sides is our first strategic move, setting the stage for further isolation. This action maintains the equation's equilibrium while bringing us closer to our goal. The result, -3y = -2x - 9, is a testament to the power of algebraic manipulation, a stepping stone towards the final form.
But the y term is not yet fully free; it's still bound by the coefficient -3. To liberate y completely, we must divide both sides of the equation by -3. This division is not just an arithmetic operation; it's a transformative act, unraveling the coefficient's grip and revealing the true nature of y. The equation gracefully morphs into y = (2/3)x + 3, a masterpiece of algebraic transformation. We have successfully navigated the complexities of the equation, arriving at the slope-intercept form. The slope, 2/3, now stands proudly, indicating the line's upward slant. The y-intercept, 3, shines brightly, marking the point where the line intersects the y-axis. The equation, once a mystery, now unveils its secrets, ready to be graphed, analyzed, and understood.
Step-by-Step Transformation of 2x-3y=-9 into Slope-Intercept Form
Let's delve into the step-by-step process of transforming the equation 2x - 3y = -9 into its slope-intercept form, a journey through algebraic manipulation that reveals the line's inherent characteristics. Each step is a deliberate move, a calculated action that brings us closer to our goal: the coveted y = mx + b form.
-
Isolating the y-term: Our initial focus is on freeing the y term from the clutches of other elements. The equation 2x - 3y = -9 presents a challenge, with both x and y intertwined. To isolate y, we employ the principle of inverse operations, the cornerstone of algebraic manipulation. Subtracting 2x from both sides of the equation is our strategic move, a deliberate action that maintains the equation's balance while shifting the focus towards y. This subtraction is not merely a mechanical step; it's a calculated maneuver that sets the stage for further isolation. The equation transforms into -3y = -2x - 9, a testament to the power of algebraic manipulation. The y term, though still encumbered by its coefficient, is now one step closer to freedom.
-
Dividing to isolate y: The y term, though partially isolated, remains tethered by the coefficient -3. To fully liberate y, we must confront this coefficient head-on. Division, the inverse operation of multiplication, is our weapon of choice. Dividing both sides of the equation -3y = -2x - 9 by -3 is not just an arithmetic operation; it's a transformative act, unraveling the coefficient's grip and revealing the true nature of y. This division must be executed meticulously, ensuring that every term is subjected to the same operation. The negative signs, often a source of confusion, must be handled with care, for a misplaced sign can alter the equation's meaning. The equation undergoes a metamorphosis, its form shifting and clarifying as the division progresses. Each term is scrutinized, each sign carefully considered, until the final transformation is complete. The result, y = (2/3)x + 3, is a masterpiece of algebraic manipulation, a testament to the power of inverse operations. The y term stands alone, free from its coefficient, ready to reveal its secrets.
-
Simplifying the equation: As we divide each term by -3, we encounter fractions, mathematical entities that demand careful handling. The fraction -2x/-3 simplifies elegantly to (2/3)x, the negative signs canceling each other out in a harmonious dance of mathematical symmetry. The constant term, -9/-3, undergoes a similar transformation, morphing into the positive integer 3, a beacon of simplicity amidst the algebraic complexities. These simplifications are not mere cosmetic changes; they are essential refinements, ensuring that the equation is presented in its most elegant and understandable form. The slope, 2/3, now stands proudly, a testament to the line's upward slant. The y-intercept, 3, shines brightly, marking the point where the line intersects the y-axis. The equation, once a tangled web of terms, now unveils its secrets, ready to be graphed, analyzed, and understood.
Interpreting the Slope-Intercept Form of y=(2/3)x+3
The equation y = (2/3)x + 3 is more than just a mathematical expression; it's a window into the soul of a line, revealing its slope and y-intercept with crystalline clarity. This form, the coveted slope-intercept form, is a powerful tool for visualizing and understanding the behavior of lines on a coordinate plane. The slope, nestled beside the x term, whispers the line's steepness and direction. The y-intercept, standing alone as a constant, marks the point where the line intersects the vertical axis.
-
Identifying the slope: In our equation, y = (2/3)x + 3, the slope takes center stage, boldly declaring its value as 2/3. This fraction, a ratio of rise over run, paints a vivid picture of the line's inclination. For every 3 units we move horizontally, the line ascends 2 units vertically. This upward trajectory is the essence of a positive slope, a testament to the line's upward climb. The slope is not just a number; it's a measure of the line's steepness, a gauge of its inclination. A larger slope signifies a steeper ascent, while a smaller slope indicates a gentler climb. The slope, in its simplicity, encapsulates the line's direction and magnitude of change.
-
Locating the y-intercept: The y-intercept, the point where the line gracefully intersects the y-axis, holds a place of prominence in the slope-intercept form. In our equation, y = (2/3)x + 3, the y-intercept reveals itself as the constant term, 3. This value is not just a number; it's a coordinate, a specific point on the coordinate plane. The y-intercept is the point (0, 3), a place where the line crosses the vertical axis. This point is not just a location; it's a starting point, a reference from which we can trace the line's path. The y-intercept, in its simplicity, anchors the line to the coordinate plane, providing a foundation for its existence.
-
Graphing the line: With the slope and y-intercept in hand, we possess the keys to graphing the line with confidence and precision. The y-intercept, (0, 3), serves as our anchor point, a fixed location from which we can begin our journey. The slope, 2/3, acts as our guide, directing our movements across the coordinate plane. From the y-intercept, we move 3 units to the right and 2 units up, marking a new point on the line. This step-by-step progression, guided by the slope, allows us to plot multiple points, each a testament to the line's trajectory. Connecting these points with a straight line, we unveil the visual representation of the equation, a line that embodies the slope and y-intercept in its form. The graph is not just a collection of points; it's a visual story, a narrative of the line's behavior across the coordinate plane.
Practical Applications of Slope-Intercept Form
The slope-intercept form, y = mx + b, transcends the realm of pure mathematics, finding practical applications in various real-world scenarios. Its ability to clearly reveal a line's slope and y-intercept makes it a valuable tool for modeling and understanding linear relationships in diverse fields.
-
Real-world scenarios: Imagine a taxi fare that starts with a base charge and increases linearly with the distance traveled. The slope-intercept form can elegantly model this scenario, where the slope represents the cost per mile and the y-intercept represents the initial base fare. Similarly, consider the growth of a plant over time, where the slope represents the growth rate and the y-intercept represents the initial height. These real-world examples highlight the versatility of the slope-intercept form in capturing linear relationships.
-
Problem-solving: The slope-intercept form is not just a descriptive tool; it's a problem-solving powerhouse. If we know two points on a line, we can determine its equation in slope-intercept form, allowing us to predict other points on the line. This predictive power is invaluable in various applications, from forecasting financial trends to estimating project timelines.
-
Mathematical analysis: Beyond practical applications, the slope-intercept form plays a crucial role in mathematical analysis. It allows us to easily compare the slopes and y-intercepts of different lines, revealing their relationships and behaviors. Parallel lines, with their identical slopes, stand in stark contrast to perpendicular lines, whose slopes bear a negative reciprocal relationship. These comparisons, facilitated by the slope-intercept form, provide deep insights into the world of linear equations.
In conclusion, transforming the equation 2x - 3y = -9 into slope-intercept form is not merely an algebraic exercise; it's a journey of discovery, revealing the line's slope and y-intercept. This transformation empowers us to graph the line, analyze its behavior, and apply it to real-world scenarios. The slope-intercept form, y = mx + b, stands as a testament to the power and elegance of linear equations, a cornerstone of mathematics and a window into the world around us.