Finding The Intersection Point And Calculating 3α - 4β

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In the realm of mathematics, particularly in coordinate geometry, the intersection of lines is a fundamental concept. Understanding how to find the point where two lines meet is crucial for solving various problems. This article delves into the process of determining the intersection point of two given linear equations and subsequently calculating a specific expression involving the coordinates of that point. We will explore the underlying principles and apply them to a concrete example, providing a step-by-step solution and detailed explanations. This article aims to provide a comprehensive understanding of the topic, making it accessible to learners of all levels.

Consider two linear equations: 2X - 5Y + 6 = 0 and 3X - 8Y + 11 = 0. These equations represent straight lines in a two-dimensional coordinate plane. The question at hand is to find the point P(α, β) where these lines intersect. Once we determine the coordinates α and β, we need to calculate the value of the expression 3α - 4β. This problem combines the concepts of linear equations, simultaneous equations, and coordinate geometry. Solving it requires a methodical approach, starting with finding the intersection point and then substituting the coordinates into the given expression. The ability to solve such problems is essential for anyone studying algebra and coordinate geometry, as it demonstrates a solid understanding of the relationship between equations and their graphical representations.

To find the intersection point P(α, β) of the lines represented by the equations 2X - 5Y + 6 = 0 and 3X - 8Y + 11 = 0, we need to solve these equations simultaneously. This means finding the values of X and Y that satisfy both equations. There are several methods to solve simultaneous equations, including substitution, elimination, and matrix methods. In this case, the elimination method is a straightforward approach. The goal is to eliminate one variable (either X or Y) by manipulating the equations so that their coefficients become equal or additive inverses. Once we eliminate one variable, we can solve for the other and then substitute the result back into one of the original equations to find the value of the eliminated variable.

Let's start by multiplying the first equation by 3 and the second equation by 2. This will make the coefficients of X in both equations equal:

First equation multiplied by 3: 3 * (2X - 5Y + 6) = 6X - 15Y + 18 = 0

Second equation multiplied by 2: 2 * (3X - 8Y + 11) = 6X - 16Y + 22 = 0

Now we have two new equations:

6X - 15Y + 18 = 0

6X - 16Y + 22 = 0

Subtract the second equation from the first equation to eliminate X:

(6X - 15Y + 18) - (6X - 16Y + 22) = 0

This simplifies to:

Y - 4 = 0

So, Y = 4. This means β = 4.

Now substitute Y = 4 into one of the original equations to solve for X. Let's use the first original equation:

2X - 5Y + 6 = 0

2X - 5(4) + 6 = 0

2X - 20 + 6 = 0

2X - 14 = 0

2X = 14

X = 7

So, X = 7. This means α = 7.

Therefore, the intersection point P(α, β) is (7, 4).

Now that we have found the intersection point P(α, β) to be (7, 4), we can proceed to calculate the value of the expression 3α - 4β. This involves substituting the values of α and β into the expression and performing the arithmetic operations.

We have α = 7 and β = 4. Substitute these values into the expression 3α - 4β:

3α - 4β = 3(7) - 4(4)

3α - 4β = 21 - 16

3α - 4β = 5

Therefore, the value of the expression 3α - 4β is 5.

In this article, we have successfully determined the intersection point of two linear equations and calculated the value of a specific expression involving its coordinates. The key steps involved solving the equations simultaneously to find the coordinates of the intersection point and then substituting those coordinates into the expression. This process demonstrates the application of fundamental concepts in algebra and coordinate geometry. The result obtained, 3α - 4β = 5, matches one of the options provided in the problem statement.

  1. Write down the given equations: 2X - 5Y + 6 = 0 and 3X - 8Y + 11 = 0.
  2. Multiply the equations by suitable constants to make the coefficients of one variable the same (or additive inverses).
  3. Subtract the equations to eliminate one variable.
  4. Solve for the remaining variable.
  5. Substitute the value obtained in step 4 into one of the original equations to solve for the other variable.
  6. Identify the intersection point P(α, β).
  7. Substitute the values of α and β into the expression 3α - 4β.
  8. Calculate the value of the expression.

We have found that the value of 3α - 4β is 5. Let's compare this with the options provided:

A. 5

B. 4

C. 2

D. 1

The correct answer is option A, which is 5. This confirms our calculations and reinforces the solution obtained through the steps outlined above.

Understanding the graphical interpretation of linear equations is crucial in coordinate geometry. Each linear equation represents a straight line, and the intersection point of two lines corresponds to the solution of the system of equations formed by those lines. In cases where lines are parallel, they do not intersect, and the system of equations has no solution. If the lines are coincident (i.e., they are the same line), they intersect at every point, and the system has infinitely many solutions. The problem we addressed in this article deals with the case where the lines intersect at exactly one point, which is a common scenario in linear algebra and coordinate geometry.

While we used the elimination method to solve the simultaneous equations, other methods can also be applied. The substitution method involves solving one equation for one variable and substituting that expression into the other equation. This method is particularly useful when one of the equations can be easily solved for one variable in terms of the other. Matrix methods, such as using determinants or inverse matrices, are also powerful tools for solving systems of linear equations, especially when dealing with larger systems involving more variables and equations. Each method has its advantages and disadvantages, and the choice of method often depends on the specific problem and personal preference.

To reinforce your understanding of the concepts discussed in this article, consider solving similar problems. For instance, you can try finding the intersection points of different pairs of linear equations and calculating expressions involving the coordinates of those points. Varying the coefficients and constants in the equations can provide additional practice and challenge your problem-solving skills. You can also explore problems involving three or more linear equations and their intersections in three-dimensional space. Such practice will help you develop a deeper understanding of linear algebra and its applications in various fields.

Finding the intersection point of two lines and calculating expressions involving its coordinates is a fundamental skill in mathematics. This article has provided a detailed explanation of the process, along with step-by-step instructions and examples. By understanding the concepts and practicing problem-solving techniques, you can enhance your mathematical abilities and excel in related topics. The intersection point represents the solution to the system of equations, and the ability to calculate expressions involving its coordinates demonstrates a strong grasp of algebraic principles. The final answer, 3α - 4β = 5, confirms the accuracy of our calculations and reinforces the importance of methodical problem-solving in mathematics.