Finding The Y-intercept Of A Line Given Two Points And Slope
In the realm of coordinate geometry, understanding the relationship between points, lines, and their equations is fundamental. This article delves into a specific problem involving a line passing through two given points and aims to determine the y-intercept of the line. We will explore the concepts of slope-intercept form, slope calculation, and equation manipulation to arrive at the solution. Let's embark on this journey to unravel the mysteries of linear equations.
Understanding the Problem: Finding the y-intercept
The core challenge lies in finding the y-intercept (b) of a line, given that the line, named Line AB, traverses through two specific points: A(-6, 6) and B(12, 3). The equation of the line is presented in the slope-intercept form, which is a quintessential representation in linear algebra, denoted as y = mx + b. Here, m symbolizes the slope of the line, which dictates its steepness and direction, while b represents the y-intercept, the point where the line intersects the y-axis. The slope, m, is already provided as -1/6. Our mission is to calculate the value of b, thereby fully defining the line's position on the coordinate plane. The options provided for b are -6, -5, 5, and 6. This problem is a classic example of applying the principles of coordinate geometry to solve for an unknown parameter in a linear equation. To effectively tackle this, we need to deeply understand the significance of each component in the slope-intercept form and how they interact to define a line.
The Slope-Intercept Form: A Deep Dive
The slope-intercept form, y = mx + b, is more than just a mathematical equation; it's a powerful tool that encapsulates the essence of a straight line. Let's dissect this equation to fully appreciate its components. The y and x are variables that represent the coordinates of any point on the line. The m, as mentioned earlier, is the slope, which quantifies how much the y-value changes for every unit change in the x-value. A negative slope, like the -1/6 in our problem, indicates that the line slopes downwards from left to right. The absolute value of the slope signifies the steepness; a larger absolute value means a steeper line. Now, the b, the y-intercept, is the pièce de résistance of our quest. It's the value of y when x is zero, essentially the point where the line cuts across the y-axis. Visualizing this on a graph, the y-intercept is where the line starts its journey, influencing the entire position of the line in the coordinate plane. To find b, we leverage the fact that any point on the line must satisfy the equation. This means we can substitute the coordinates of either point A or point B into the equation and solve for b. This is a fundamental concept in coordinate geometry, where points and equations are intertwined, each informing the other.
Calculating the y-intercept: A Step-by-Step Approach
With a firm grasp of the slope-intercept form, we can now systematically calculate the y-intercept (b). We are given two points, A(-6, 6) and B(12, 3), and the slope m = -1/6. The strategy here is to substitute the coordinates of either point into the equation y = mx + b and then solve for b. Let’s start with point A(-6, 6). Substituting x = -6 and y = 6, and m = -1/6 into the equation, we get:
6 = (-1/6) * (-6) + b
This simplifies to:
6 = 1 + b
Subtracting 1 from both sides, we find:
b = 5
Now, let's verify this by using point B(12, 3). Substituting x = 12 and y = 3, and m = -1/6 into the equation, we get:
3 = (-1/6) * (12) + b
This simplifies to:
3 = -2 + b
Adding 2 to both sides, we again find:
b = 5
Both points A and B yield the same value for b, which is 5. This consistency confirms the accuracy of our calculation. The y-intercept, therefore, is 5, meaning the line intersects the y-axis at the point (0, 5). This methodical approach, using substitution and algebraic manipulation, is a cornerstone of solving linear equations in coordinate geometry. By understanding this process, we can confidently tackle similar problems involving lines and their equations.
Verifying the Solution: Ensuring Accuracy
After calculating the y-intercept (b) to be 5, it's crucial to verify the solution. Verification ensures the accuracy of our calculations and reinforces our understanding of the problem. We've already used both points A and B to calculate b, but we can further verify by constructing the full equation of the line and checking if it aligns with the given information. The equation of the line is now y = (-1/6)x + 5. To verify, we can substitute the x-coordinates of points A and B into this equation and see if we get the corresponding y-coordinates.
For point A(-6, 6):
y = (-1/6) * (-6) + 5
y = 1 + 5
y = 6
This matches the y-coordinate of point A, confirming that point A lies on the line.
For point B(12, 3):
y = (-1/6) * (12) + 5
y = -2 + 5
y = 3
This matches the y-coordinate of point B, further solidifying that point B also lies on the line. An additional way to verify is to visualize the line on a graph. Plotting points A and B and drawing a line through them, we can visually confirm that the line indeed intersects the y-axis at 5. This verification step is not just about confirming the answer; it’s about reinforcing the concepts and building confidence in our problem-solving abilities. It highlights the interconnectedness of points, slopes, and intercepts in defining a line.
Conclusion: The Significance of the y-intercept
In conclusion, by systematically applying the principles of coordinate geometry, we successfully determined the y-intercept (b) of Line AB to be 5. We started by understanding the problem, which involved finding b given two points and the slope of the line. We then delved into the slope-intercept form, dissecting its components and understanding how they define a line. The core of the solution involved substituting the coordinates of the given points into the slope-intercept equation and solving for b. We meticulously verified our solution using both points and by constructing the full equation of the line. This exercise underscores the significance of the y-intercept in defining a line. The y-intercept, as the point where the line intersects the y-axis, plays a crucial role in positioning the line on the coordinate plane. It, along with the slope, uniquely defines a straight line. Understanding how to calculate and interpret the y-intercept is a fundamental skill in mathematics, with applications extending beyond textbook problems to real-world scenarios involving linear relationships. This problem serves as a testament to the power of algebraic manipulation and the importance of verification in ensuring the accuracy of solutions. The y-intercept is not just a number; it’s a key piece of information that unlocks the full picture of a linear equation.
Answer:
The value of b is 5.
Keywords
Slope-intercept form, y-intercept, linear equations, coordinate geometry, slope, points, equation of a line, algebraic manipulation, solution verification
