Finding The Value Of Q For A Specific Solution Set In A Linear System
In the realm of linear algebra, solving systems of equations is a fundamental skill. These systems often represent real-world scenarios, from balancing chemical equations to optimizing resource allocation. One interesting challenge is determining the conditions under which a system has infinitely many solutions, all lying on a specific line. In this article, we delve into such a problem, exploring how to find the value of a parameter that makes the solution set of a linear system a given line. Specifically, we aim to find the value of Q in the following system of linear equations so that the solution to the system is the set of all points (x, y) that satisfy the equation x - 3y = 4:
x - 3y = 4
2x + Qy = 8
This problem combines algebraic manipulation with a geometric understanding of linear equations and their solutions. By analyzing the relationships between the equations, we can determine the value of Q that ensures the two lines coincide, leading to infinitely many solutions along the given line. This concept is crucial in various fields, including engineering, economics, and computer science, where systems of equations are frequently used to model and solve problems.
To effectively solve this problem, it's crucial to grasp the underlying concepts. A system of linear equations represents a set of lines in a coordinate plane. The solution to the system is the set of points where these lines intersect. There are three possible scenarios:
- The lines intersect at a single point, giving a unique solution.
- The lines are parallel and never intersect, resulting in no solution.
- The lines coincide, meaning they are the same line, leading to infinitely many solutions. These solutions lie on the common line.
In our case, we are given that the solution set is the line x - 3y = 4. This means that the second equation, 2x + Qy = 8, must represent the same line. For two linear equations to represent the same line, they must be scalar multiples of each other. In other words, one equation can be obtained by multiplying the other equation by a constant.
This understanding forms the basis of our approach. We will manipulate the given equations to find the value of Q that makes the second equation a multiple of the first. This will ensure that the two lines coincide, and the solution set is indeed the given line x - 3y = 4. This process involves algebraic manipulation and a careful comparison of coefficients.
Our goal is to find the value of Q such that the equation 2x + Qy = 8 represents the same line as x - 3y = 4. To achieve this, we need to make the second equation a multiple of the first. Observe that the x-coefficient in the second equation is 2, which is twice the x-coefficient in the first equation (which is 1). This suggests that we should multiply the entire first equation by 2:
2 * (x - 3y) = 2 * 4
This simplifies to:
2x - 6y = 8
Now, we have two equations:
2x - 6y = 8
2x + Qy = 8
For these two equations to represent the same line, their coefficients must be equal. Comparing the coefficients of y, we have:
Q = -6
Therefore, the value of Q that makes the solution set of the system the line x - 3y = 4 is -6. This means that when Q is -6, the second equation becomes 2x - 6y = 8, which is simply twice the first equation. Thus, the two equations are equivalent, and their graphs coincide, resulting in infinitely many solutions along the line x - 3y = 4. This result confirms our initial approach of finding a scalar multiple relationship between the equations.
To ensure our solution is correct, we can substitute Q = -6 back into the original system and verify that the solution set is indeed x - 3y = 4. With Q = -6, the system becomes:
x - 3y = 4
2x - 6y = 8
We can see that the second equation is simply twice the first equation. This confirms that the two equations represent the same line. To further verify, we can try to solve the system using methods like substitution or elimination. If we multiply the first equation by 2, we get 2x - 6y = 8, which is identical to the second equation. This means that the equations are dependent, and there are infinitely many solutions.
To express the solution set, we can solve the first equation for x:
x = 3y + 4
This equation represents the line x - 3y = 4. Any point (x, y) that satisfies this equation is a solution to the system. For example, if y = 0, then x = 4, and (4, 0) is a solution. If y = 1, then x = 7, and (7, 1) is a solution. The set of all such points forms the solution set, which is the line x - 3y = 4. This verification step is crucial to ensure the accuracy of our result and to solidify our understanding of the problem.
In this article, we have successfully determined the value of Q that makes the solution set of the given linear system the line x - 3y = 4. By recognizing that the two equations must represent the same line for this condition to hold, we were able to find the value Q = -6. This involved manipulating the equations to find a scalar multiple relationship and comparing coefficients. We then verified our solution by substituting Q = -6 back into the system and confirming that the equations are dependent and represent the same line.
This problem highlights the importance of understanding the geometric interpretation of linear systems. The concept of lines coinciding and resulting in infinitely many solutions is fundamental in linear algebra and has applications in various fields. The ability to solve such problems demonstrates a strong grasp of algebraic manipulation and problem-solving skills. Moreover, this exercise reinforces the importance of verification in mathematics, ensuring the accuracy and reliability of our results. By mastering these techniques, students and practitioners can confidently tackle more complex problems involving linear systems and their solutions. The skills and concepts discussed here are essential building blocks for advanced topics in mathematics, engineering, and other quantitative disciplines.
