Points On A Line And Collinearity Exploring Solutions And Concepts

Decoding the Line 2x + y = 6: A Quest for Points of Intersection

In the realm of coordinate geometry, the equation 2x + y = 6 represents a straight line that stretches infinitely across the Cartesian plane. Our mission is to identify which of the given points – (2, 0), (0, 6), and (1, 4) – gracefully reside on this line. To accomplish this, we embark on a journey of substitution, where we meticulously replace the variables x and y in the equation with the corresponding coordinates of each point.

Let's begin with the point (2, 0). Substituting x = 2 and y = 0 into the equation 2x + y = 6, we arrive at 2(2) + 0 = 4, which emphatically does not equal 6. Therefore, the point (2, 0) stands apart from the line, failing to satisfy its equation.

Next, we turn our attention to the point (0, 6). With x = 0 and y = 6, the equation 2x + y = 6 transforms into 2(0) + 6 = 6, a resounding affirmation that the point (0, 6) proudly occupies a position on the line. Its coordinates seamlessly align with the equation, confirming its membership.

Finally, we examine the point (1, 4). Substituting x = 1 and y = 4 into the equation 2x + y = 6, we obtain 2(1) + 4 = 6, yet another harmonious match. The point (1, 4) joins the ranks of (0, 6), firmly establishing itself on the line 2x + y = 6.

Thus, our investigation reveals that points (0, 6) and (1, 4) are the chosen ones, the points that gracefully lie on the line 2x + y = 6. The correct answer, therefore, is option B, which proudly proclaims that 2 and 3 only are the true inhabitants of this linear realm.

Delving Deeper: The Significance of Points on a Line

Understanding the concept of points lying on a line is fundamental in coordinate geometry. It allows us to visualize and interpret linear equations, and it forms the basis for numerous applications in mathematics, science, and engineering. The ability to identify points on a line is crucial for graphing linear equations, solving systems of equations, and modeling real-world phenomena that exhibit linear relationships. Consider, for instance, the relationship between distance, speed, and time in uniform motion. This relationship can be represented by a linear equation, and identifying points on the line allows us to determine the distance traveled at various times.

Beyond the Equation: Visualizing the Line

The equation 2x + y = 6 can be visualized as a straight line on a graph. The points that satisfy the equation are precisely those that lie on this line. This visual representation provides a powerful way to understand the relationship between the equation and its solutions. By plotting the points (0, 6) and (1, 4) on a graph, we can draw a line that passes through both points, confirming that these points indeed lie on the line represented by the equation 2x + y = 6.

Unraveling the Mystery of Collinear Points: Finding the Elusive Value of k

Our next quest takes us into the realm of collinearity, where points align in perfect harmony along a single straight line. We are presented with three points – (9, k), (3, 2), and (1, 1) – and tasked with uncovering the value of k that binds them together on the same linear path.

Collinear points share a unique characteristic: the slopes between any two pairs of points among them are always equal. This fundamental property serves as our guiding principle in unraveling the mystery of k. To begin, we calculate the slope between the points (3, 2) and (1, 1), employing the slope formula: (y2 - y1) / (x2 - x1). Substituting the coordinates, we obtain (1 - 2) / (1 - 3) = (-1) / (-2) = 1/2. This value represents the slope of the line connecting these two points.

Now, we turn our attention to the slope between the points (9, k) and (3, 2). Again, we invoke the slope formula, arriving at (k - 2) / (9 - 3) = (k - 2) / 6. For these three points to be collinear, this slope must be equal to the slope we previously calculated, 1/2. Thus, we set up the equation (k - 2) / 6 = 1/2 and embark on a journey of algebraic manipulation to isolate k.

Multiplying both sides of the equation by 6, we obtain k - 2 = 3. Adding 2 to both sides, we triumphantly arrive at k = 5. This is the elusive value of k that ensures the points (9, k), (3, 2), and (1, 1) march in unison along the same straight line.

Therefore, the correct answer is not among the options provided (A. 4 or B). The correct value of k is 5.

The Essence of Collinearity: A Symphony of Slopes

The concept of collinearity is essential in geometry, offering a powerful tool for determining if points lie on the same line. This concept has practical applications in various fields, including surveying, navigation, and computer graphics. For example, in surveying, collinearity is used to ensure that points are aligned along a straight line, which is crucial for accurate measurements and mapping. In computer graphics, collinearity is used to determine if points lie on the same line segment, which is essential for rendering lines and shapes.

Beyond the Slope: Alternative Approaches to Collinearity

While the slope method is a common approach to determining collinearity, alternative methods exist. One such method involves calculating the area of the triangle formed by the three points. If the area is zero, the points are collinear. This method leverages the fact that a triangle with zero area degenerates into a line segment. Another method involves using determinants, a concept from linear algebra. If the determinant of a matrix formed by the coordinates of the points is zero, the points are collinear. These alternative methods provide different perspectives on the concept of collinearity and can be useful in various situations.

The Power of Visualization: Collinear Points on a Graph

Collinear points can be visualized as points that lie on the same straight line on a graph. This visual representation provides an intuitive understanding of the concept of collinearity. By plotting the points (9, 5), (3, 2), and (1, 1) on a graph, we can observe that they indeed lie on the same line, confirming our calculated value of k = 5. The visual confirmation reinforces the analytical solution, providing a comprehensive understanding of the problem.

In summary, understanding the concepts of points on a line and collinearity is crucial for mastering coordinate geometry. These concepts are not only fundamental in mathematics but also have wide-ranging applications in various fields. By mastering these concepts, we unlock a deeper understanding of the world around us and gain the ability to solve complex problems with elegance and precision. Whether it's determining if a point lies on a line or finding the value that aligns points in perfect collinearity, the principles of coordinate geometry empower us to explore and unravel the mysteries of the geometric realm. This exploration, filled with the joy of discovery, ultimately enhances our mathematical prowess and broadens our intellectual horizons. The journey through the world of coordinate geometry is a testament to the beauty and power of mathematical reasoning, a journey that continues to inspire and challenge us to seek further understanding.