Solving Y=2(5x+1)+3(5x+3) Expressing In Slope-Intercept Form
In the realm of mathematics, linear equations reign supreme, forming the bedrock of countless concepts and applications. Among the various forms of linear equations, the slope-intercept form stands out for its clarity and ease of interpretation. This form, expressed as y = mx + b, unveils the very essence of a line: its slope (m) and its y-intercept (b). In this comprehensive guide, we embark on a journey to master the art of transforming equations into slope-intercept form, equipping you with the skills to dissect and decipher linear relationships.
Understanding the Slope-Intercept Form
At its core, the slope-intercept form, y = mx + b, presents a linear equation in its most revealing light. The coefficient m gracefully unveils the line's slope, dictating its steepness and direction. A positive m signifies an upward trajectory, while a negative m indicates a downward descent. The magnitude of m further quantifies the steepness, with larger absolute values implying steeper inclines. The constant term b, on the other hand, gracefully marks the y-intercept, the point where the line gracefully intersects the vertical axis. This dynamic duo, the slope and y-intercept, collectively paints a vivid portrait of the line's behavior and positioning within the Cartesian plane.
The Art of Transformation: Converting Equations to Slope-Intercept Form
While the slope-intercept form offers unparalleled clarity, linear equations often appear in disguise, cloaked in alternative forms. The general form, Ax + By = C, and the point-slope form, y - y1 = m(x - x1), represent but a few of the disguises linear equations may assume. Fear not, for we shall embark on a transformative journey, mastering the art of converting these disguised equations into the revealing slope-intercept form. The key lies in strategically employing algebraic manipulations, skillfully isolating y on one side of the equation, while gracefully arranging the remaining terms in the coveted mx + b format. Let's delve into the intricacies of this transformative process, unveiling the secrets to unlocking the slope-intercept form from any linear equation.
Illustrative Example: Unveiling the Slope-Intercept Form
Let's consider the equation y = 2(5x + 1) + 3(5x + 3), a seemingly complex expression yearning for simplification. Our mission is to transform this equation into the elegant slope-intercept form, y = mx + b. To achieve this, we shall embark on a step-by-step odyssey, employing the power of algebraic manipulation.
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The Distributive Property: Our journey commences with the distributive property, a fundamental principle that empowers us to unravel expressions nestled within parentheses. By distributing the coefficients, we liberate the terms from their enclosed confines, paving the way for simplification. Applying the distributive property to our equation, we gracefully expand the expression:
y = 2(5x + 1) + 3(5x + 3)
y = 10x + 2 + 15x + 9
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Combining Like Terms: With the parentheses vanquished, our attention now turns to the realm of like terms, terms that share the same variable and exponent. Like terms, much like kindred spirits, yearn to be united, their coefficients coalescing to form a single, harmonious entity. In our equation, 10x and 15x stand as like terms, as do the constants 2 and 9. By meticulously combining these like terms, we simplify our equation:
y = 10x + 2 + 15x + 9
y = (10x + 15x) + (2 + 9)
y = 25x + 11
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Slope-Intercept Form Achieved: Behold, the culmination of our transformative journey! Through the strategic application of the distributive property and the meticulous combination of like terms, we have successfully transformed our equation into the coveted slope-intercept form:
y = 25x + 11
In this triumphant form, the slope, m, stands proudly as 25, signifying a steep upward incline. The y-intercept, b, graces the equation as 11, marking the point where the line gracefully intersects the vertical axis. Our mission is accomplished, the slope-intercept form unveiled, and the secrets of the line revealed.
Delving Deeper: Unveiling the Slope and Y-Intercept
With our equation now elegantly expressed in slope-intercept form, y = 25x + 11, we are poised to dissect its essence, extracting the fundamental properties that define its behavior. The coefficient of x, 25, proudly proclaims itself as the slope, m. This numerical beacon illuminates the line's steepness and direction. A slope of 25 signifies a rapid ascent, for every unit increase in x, y surges upwards by 25 units. The positive sign further solidifies the upward trajectory, confirming that the line gracefully climbs as we traverse from left to right.
The constant term, 11, gracefully assumes the mantle of the y-intercept, b. This numerical landmark marks the precise point where the line intersects the vertical axis. At the point (0, 11), the line gracefully kisses the y-axis, forever etching its presence in the Cartesian plane.
Practical Applications: The Power of Slope-Intercept Form
The slope-intercept form transcends the realm of pure mathematical abstraction, finding practical applications in a myriad of real-world scenarios. Its ability to succinctly encapsulate the essence of a line makes it an invaluable tool for modeling and analyzing linear relationships.
Imagine, for instance, a scenario where you're charting the trajectory of a hot air balloon. The balloon's ascent can be elegantly modeled using a linear equation, with the slope representing the rate of ascent and the y-intercept signifying the initial altitude. By expressing this equation in slope-intercept form, you gain immediate insights into the balloon's behavior, readily predicting its altitude at any given time.
In the realm of economics, the slope-intercept form finds its niche in modeling cost functions. A linear cost function, expressed in slope-intercept form, unveils the relationship between production costs and the number of units produced. The slope embodies the variable cost per unit, while the y-intercept represents the fixed costs, the expenses incurred regardless of production volume. This knowledge empowers businesses to make informed decisions, optimizing production strategies and maximizing profitability.
The slope-intercept form extends its reach to the realm of physics, where it elegantly describes motion at a constant velocity. The equation d = vt + d0, where d represents distance, v signifies velocity, t denotes time, and d0 marks the initial distance, mirrors the slope-intercept form. Here, the velocity gracefully assumes the role of the slope, dictating the rate of change in distance, while the initial distance acts as the y-intercept, marking the starting point of the motion.
These examples merely scratch the surface of the slope-intercept form's practical prowess. From charting financial growth to modeling physical phenomena, this versatile form empowers us to understand and predict linear relationships across a vast spectrum of disciplines.
Mastering the Art: A Recap of the Slope-Intercept Form
As we draw the curtains on our exploration of the slope-intercept form, let's recap the key takeaways that will solidify your mastery of this fundamental concept.
The slope-intercept form, y = mx + b, stands as a beacon of clarity, revealing the essence of a line through its slope (m) and y-intercept (b). The slope, m, dictates the line's steepness and direction, while the y-intercept, b, marks the point where the line gracefully intersects the vertical axis.
To transform equations into slope-intercept form, we employ the power of algebraic manipulation, strategically isolating y on one side of the equation, while gracefully arranging the remaining terms in the coveted mx + b format. The distributive property and the meticulous combination of like terms serve as our trusty tools in this transformative endeavor.
The slope-intercept form transcends theoretical musings, finding practical applications in a myriad of real-world scenarios. From charting trajectories to modeling economic behavior, this versatile form empowers us to understand and predict linear relationships across a vast spectrum of disciplines.
With these principles firmly etched in your mind, you are now equipped to confidently wield the slope-intercept form, unraveling the mysteries of linear equations and harnessing their power to illuminate the world around you.
Therefore, the correct answer is A. y=25x+11
