Alejandra's Line Transformation Finding The Equation Of A New Line
Introduction
In the realm of mathematics, linear equations serve as fundamental tools for representing and analyzing straight lines. These equations, often expressed in various forms such as slope-intercept or point-slope, provide a concise way to describe the relationship between two variables and the characteristics of the line they define. In this article, we delve into a problem involving a linear equation and its transformation. We'll explore how changes in the slope of a line, while keeping a specific point constant, affect its equation. This problem, presented in the context of a student named Alejandra and her teacher, offers a practical application of linear equation concepts and provides insights into the interplay between geometric and algebraic representations of lines. Let's embark on this mathematical journey, unraveling the intricacies of linear equations and their transformations.
Understanding the Initial Equation
The problem begins with Alejandra, who has correctly written the equation
y - 3 = (1/5)(x - 10)
to represent a line sketched by her teacher. This equation is in point-slope form, a particularly useful format that directly reveals both the slope of the line and a point it passes through. The point-slope form is generally expressed as:
y - y₁ = m(x - x₁)
where m represents the slope of the line, and (x₁, y₁) denotes a specific point on the line. By comparing Alejandra's equation to the general point-slope form, we can readily identify the slope and a point on the line. In this case, the slope, m, is 1/5, and the point (x₁, y₁) is (10, 3). This means that the original line has a gentle upward slant (a slope of 1/5) and passes directly through the point with coordinates x=10 and y=3 on the Cartesian plane. This foundational understanding of the initial equation is crucial for comprehending the subsequent transformation and finding the equation of the new line.
The Teacher's Transformation: Changing the Slope
The core of the problem lies in the transformation introduced by Alejandra's teacher. The teacher modifies the line in a specific way: the slope is changed to 2, but the line is still constrained to pass through the same point. This is a critical piece of information, as it provides the link between the original line and the transformed line. The point (10, 3) acts as an anchor, ensuring that the new line, despite having a different steepness (a slope of 2 instead of 1/5), maintains a connection to the original line. The change in slope signifies a significant alteration in the line's inclination. A slope of 2 indicates a steeper upward slant compared to the original slope of 1/5. This transformation highlights the fundamental role of the slope in determining a line's direction and steepness. To mathematically represent this new line, we need to construct an equation that reflects both the new slope and the fixed point. This is where our understanding of point-slope form becomes invaluable.
Finding the New Equation: Applying Point-Slope Form
To determine the equation of the transformed line, we leverage the point-slope form, which, as mentioned earlier, is expressed as:
y - y₁ = m(x - x₁)
We already know the crucial pieces of information needed to construct this equation: the new slope, m = 2, and the point the line passes through, (x₁, y₁) = (10, 3). Substituting these values into the point-slope form, we get:
y - 3 = 2(x - 10)
This equation succinctly represents the transformed line. It captures the essence of the teacher's modification: a line with a slope of 2 passing through the point (10, 3). This equation is the answer to the problem. It mathematically describes the line that Alejandra needs to represent after the teacher's transformation. However, while this point-slope form is perfectly valid, it's often beneficial to explore other forms of linear equations, such as slope-intercept form, to gain a more comprehensive understanding of the line's characteristics. The ability to convert between different forms of linear equations is a valuable skill in mathematics, allowing for flexibility in analysis and problem-solving.
Converting to Slope-Intercept Form (Optional)
While the point-slope form y - 3 = 2(x - 10) is a correct representation of the transformed line, we can further manipulate it to express the equation in slope-intercept form. The slope-intercept form is written as:
y = mx + b
where m is the slope and b is the y-intercept (the point where the line crosses the y-axis). To convert our point-slope equation to slope-intercept form, we need to isolate y on one side of the equation. We begin by distributing the 2 on the right side:
y - 3 = 2x - 20
Next, we add 3 to both sides to isolate y:
y = 2x - 17
This is the slope-intercept form of the equation. It clearly shows that the slope is 2 (as we already knew) and that the y-intercept is -17. This means the line crosses the y-axis at the point (0, -17). Converting to slope-intercept form provides additional insight into the line's behavior and its relationship to the coordinate axes. It's a valuable exercise in algebraic manipulation and reinforces the connection between different representations of linear equations. However, for the original problem, the point-slope form y - 3 = 2(x - 10) is a perfectly acceptable and correct answer.
Conclusion
This problem illustrates the power and versatility of linear equations in representing and manipulating lines. By understanding the point-slope form, we were able to readily construct the equation of the transformed line after the teacher changed the slope. The key was to recognize that the point (10, 3) remained constant, acting as a pivot around which the line rotated as its slope changed. Furthermore, we explored the conversion to slope-intercept form, demonstrating how different forms of linear equations can provide complementary insights into a line's characteristics. This exercise underscores the importance of mastering linear equations, as they form the foundation for more advanced mathematical concepts and have wide-ranging applications in various fields. The ability to manipulate and interpret linear equations is a crucial skill for anyone pursuing further studies in mathematics, science, or engineering. Alejandra's problem, though seemingly simple, encapsulates fundamental principles that are essential for mathematical proficiency. In essence, this problem highlights that by mastering the various forms of linear equations, especially the point-slope form, and their conversions, we can easily solve mathematical problems. The core idea was understanding how changing the slope affects the line equation while keeping a specific point constant, serving as a practical application of linear equation concepts. Linear equations, therefore, become more than just mathematical expressions; they are tools for exploring and describing the world around us, making them a cornerstone of mathematical literacy and problem-solving.
What equation should Alejandra write to represent the new line after the teacher changed its slope to 2, while still passing through the same point as the original line?
Alejandra's Line Transformation Finding the Equation of a New Line
