Finding Intersection Point Of 3x - Y - 5 = 0 And 12x + Y - 25 = 0

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In the realm of mathematics, particularly in coordinate geometry, finding the intersection point of two loci is a fundamental concept with wide-ranging applications. Loci, the plural of locus, refer to the set of all points that satisfy a given condition or equation. When we have two such conditions or equations, their intersection represents the point(s) that simultaneously satisfy both. This article delves into the process of finding the intersection point of two loci, using the specific example of two linear equations: 3x - y - 5 = 0 and 12x + y - 25 = 0. We will explore the underlying principles, various methods for solving such problems, and the significance of these solutions in different contexts. Understanding how to determine the intersection of loci is crucial for various mathematical and real-world applications, including solving systems of equations, graphical analysis, and optimization problems. This article aims to provide a comprehensive guide to this concept, making it accessible to students and enthusiasts alike.

To fully grasp the concept of finding the intersection point, it's essential to first understand what loci are. A locus (plural: loci) is essentially a set of points that satisfy a specific condition or equation. Imagine a line drawn on a graph; each point on that line adheres to the equation that defines the line. This line is a locus. Similarly, a circle is a locus of points equidistant from a center point. When we have two loci, their intersection points are the points that exist on both loci simultaneously. In other words, these points satisfy the equations or conditions of both loci. Consider two straight lines on a coordinate plane. The point where they cross each other is the intersection point, a unique location that lies on both lines. This point's coordinates, (x, y), will satisfy both equations representing the lines. The concept of intersecting loci extends beyond straight lines. It can apply to circles, parabolas, and other geometric shapes, making it a versatile tool in mathematical analysis. Understanding this fundamental concept is the key to solving more complex problems in coordinate geometry and beyond.

Our specific problem involves finding the point of intersection for two linear equations:

  1. 3x - y - 5 = 0
  2. 12x + y - 25 = 0

These equations represent two straight lines on a Cartesian plane. Each equation defines a set of points (x, y) that, when plotted, form a straight line. The point of intersection, if it exists, is the unique coordinate (x, y) that satisfies both equations simultaneously. Geometrically, this is where the two lines cross each other on the graph. To find this point, we need to solve the system of equations. This can be done using various algebraic methods, including substitution, elimination, or matrix methods. In this article, we will focus on the elimination and substitution methods, as they are particularly effective for solving systems of linear equations. By solving this problem, we will gain a deeper understanding of how to find the intersection points of loci represented by linear equations, a skill that is widely applicable in mathematics and related fields.

Method 1: Elimination Method

The elimination method is a powerful algebraic technique for solving systems of linear equations. The core idea behind this method is to manipulate the equations in such a way that one of the variables is eliminated when the equations are added or subtracted. This simplifies the system, allowing us to solve for the remaining variable. Once we find the value of one variable, we can substitute it back into either of the original equations to solve for the other variable. This method is particularly effective when the coefficients of one of the variables are the same or can be easily made the same by multiplying one or both equations by a constant. In our case, we can readily eliminate the 'y' variable because the coefficients of 'y' in the two equations are opposites (-1 and +1). This direct cancellation makes the elimination method a straightforward and efficient choice for this problem.

Step-by-step Solution using Elimination Method

Let's walk through the solution step-by-step using the elimination method for the given equations:

  1. Write down the equations:
    • 3x - y - 5 = 0
    • 12x + y - 25 = 0
  2. Observe that the coefficients of 'y' are -1 and +1. They are already in a form that allows for direct elimination. No multiplication is needed in this case.
  3. Add the two equations: (3x - y - 5) + (12x + y - 25) = 0 + 0
  4. Combine like terms: 15x - 30 = 0
  5. Solve for 'x':
    • Add 30 to both sides: 15x = 30
    • Divide both sides by 15: x = 2
  6. Now that we have the value of 'x', substitute it back into either of the original equations to solve for 'y'. Let's use the first equation:
    • 3(2) - y - 5 = 0
    • 6 - y - 5 = 0
    • 1 - y = 0
    • y = 1
  7. Therefore, the point of intersection is (2, 1).

This step-by-step approach clearly demonstrates the simplicity and efficiency of the elimination method. By strategically adding the equations, we eliminated 'y' and were able to solve for 'x'. Then, a simple substitution gave us the value of 'y'.

Method 2: Substitution Method

The substitution method provides an alternative approach to solving systems of linear equations. In this method, we solve one of the equations for one variable in terms of the other, and then substitute that expression into the other equation. This process results in a single equation with a single variable, which can be easily solved. Once we have the value of one variable, we substitute it back into either of the original equations (or the expression we derived earlier) to find the value of the other variable. The substitution method is particularly useful when one of the equations can be easily rearranged to isolate a variable. For instance, if one equation has a variable with a coefficient of 1, it's often straightforward to solve for that variable and substitute the resulting expression into the other equation.

Step-by-step Solution using Substitution Method

Let's apply the substitution method to our system of equations:

  1. Write down the equations:
    • 3x - y - 5 = 0
    • 12x + y - 25 = 0
  2. Solve the first equation for 'y':
    • 3x - y - 5 = 0
    • -y = -3x + 5
    • y = 3x - 5
  3. Substitute this expression for 'y' into the second equation:
    • 12x + (3x - 5) - 25 = 0
  4. Simplify and solve for 'x':
    • 12x + 3x - 5 - 25 = 0
    • 15x - 30 = 0
    • 15x = 30
    • x = 2
  5. Substitute the value of 'x' back into the expression for 'y' we found earlier:
    • y = 3(2) - 5
    • y = 6 - 5
    • y = 1
  6. Therefore, the point of intersection is (2, 1).

This step-by-step breakdown illustrates how the substitution method works. By isolating 'y' in the first equation and substituting it into the second, we created an equation with only 'x'. Solving for 'x' and then substituting back gave us the value of 'y', leading us to the point of intersection.

Once we have found a potential solution, it is crucial to verify that it satisfies both original equations. This step ensures that we have indeed found the correct point of intersection. To verify our solution (2, 1), we will substitute x = 2 and y = 1 into both equations:

  1. First Equation: 3x - y - 5 = 0
    • Substitute x = 2 and y = 1:
      • 3(2) - 1 - 5 = 0
      • 6 - 1 - 5 = 0
      • 0 = 0 (The equation holds true)
  2. Second Equation: 12x + y - 25 = 0
    • Substitute x = 2 and y = 1:
      • 12(2) + 1 - 25 = 0
      • 24 + 1 - 25 = 0
      • 0 = 0 (The equation holds true)

Since the solution (2, 1) satisfies both equations, we can confidently conclude that it is the correct point of intersection for the two given lines. Verification is an essential step in problem-solving, as it eliminates the possibility of errors and confirms the accuracy of our result. This practice is not only important in mathematics but also in various fields where precision and correctness are paramount.

A graphical representation can provide valuable insights into the concept of intersecting loci. Each of the equations, 3x - y - 5 = 0 and 12x + y - 25 = 0, represents a straight line on the Cartesian plane. When we plot these lines on a graph, the point where they intersect visually represents the solution to the system of equations. In our case, the point of intersection is (2, 1). This means that the lines cross each other at the coordinates x = 2 and y = 1. Graphing the equations not only confirms our algebraic solution but also provides a visual understanding of the problem. It's a powerful tool for conceptualizing the relationship between equations and their geometric representations. Furthermore, graphical analysis can help in identifying the nature of the solutions. For example, if the lines are parallel, they will not intersect, indicating that there is no solution to the system of equations. If the lines are coincident (i.e., they are the same line), they will intersect at every point, indicating infinitely many solutions. Thus, the graphical interpretation is a valuable complement to algebraic methods in solving systems of equations and understanding intersecting loci.

The concept of finding the point of intersection of loci is not just an abstract mathematical idea; it has numerous applications in various fields. In mathematics, it is fundamental to solving systems of equations, which arise in linear algebra, calculus, and differential equations. For instance, in linear programming, finding the intersection points of constraint equations helps determine the feasible region for optimization problems. In geometry, this concept is used to find the centers of circles, the foci of ellipses, and other geometric properties. Beyond mathematics, the application of intersecting loci extends to real-world scenarios. In physics, it can be used to calculate the trajectories of objects, such as projectiles or satellites, by finding the intersection of their paths. In engineering, it is used in the design of structures, circuits, and control systems. In economics, it can help determine equilibrium points in supply and demand models. The versatility of this concept makes it an essential tool in problem-solving across various disciplines. Understanding how to find the intersection of loci empowers individuals to tackle a wide range of challenges in both theoretical and practical contexts.

In conclusion, finding the point of intersection of two loci is a fundamental concept in mathematics with broad applications. In this article, we specifically addressed the problem of finding the intersection of two linear equations: 3x - y - 5 = 0 and 12x + y - 25 = 0. We explored two primary methods for solving this problem: the elimination method and the substitution method. Both methods led us to the same solution, the point (2, 1), which represents the coordinates where the two lines intersect on the Cartesian plane. We also emphasized the importance of verifying the solution by substituting the values back into the original equations, ensuring accuracy. Furthermore, we discussed the graphical interpretation of intersecting loci, highlighting how visualizing the lines on a graph can provide a deeper understanding of the problem and its solution. Finally, we touched upon the diverse applications of this concept in various fields, from mathematics and physics to engineering and economics. Mastering the technique of finding the intersection of loci is a valuable skill that equips individuals with the tools to solve a wide range of problems in both academic and real-world contexts. Whether it's solving systems of equations, optimizing designs, or modeling real-world phenomena, this fundamental concept serves as a cornerstone for analytical and problem-solving endeavors.