Finding The Value Of Q In A System Of Equations For Infinite Solutions

In the realm of mathematics, systems of linear equations play a crucial role. A system of linear equations is a set of two or more linear equations containing the same variables. The solution to a system of linear equations is the set of values that, when substituted for the variables, make all the equations true. Systems of equations can have one solution, no solution, or infinitely many solutions. This article delves into a specific scenario where we aim to find the value of a constant, denoted as Q, in a system of two linear equations such that the solution set consists of all points lying on a given line. This means the system has infinitely many solutions, and the two lines represented by the equations are essentially the same. We will explore the conditions for this to occur and the steps involved in determining the value of Q.

We are given the following system of linear equations:

$$\begin{aligned} x - 3y &= 4 \\ 2x - 6y &= Q \end{aligned}$$

The task is to find the value of Q such that the solution to this system is the set of all points (x, y) that satisfy the equation x - 3y = 4. In other words, we want the system to have infinitely many solutions, and these solutions must lie on the line defined by the equation x - 3y = 4. This implies that the second equation, 2x - 6y = Q, must be a multiple of the first equation. Understanding the relationship between the equations is key to solving this problem. When one equation is a multiple of another, they represent the same line, leading to an infinite number of solutions.

For a system of two linear equations in two variables to have infinitely many solutions, the two equations must represent the same line. This means that one equation is a scalar multiple of the other. In other words, if we multiply the first equation by a constant, we should obtain the second equation. This condition ensures that the two equations are dependent, and their graphs coincide, resulting in an infinite number of points of intersection.

Consider the general form of two linear equations:

$$\begin{aligned} a_1x + b_1y &= c_1 \\ a_2x + b_2y &= c_2 \end{aligned}$$

For these equations to represent the same line, the ratios of their coefficients must be equal. That is:

$$\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$$

In our specific case, this condition will help us determine the value of Q that makes the second equation a multiple of the first, ensuring infinitely many solutions.

To find the value of Q, we need to examine the given system of equations:

$$\begin{aligned} x - 3y &= 4 \\ 2x - 6y &= Q \end{aligned}$$

Notice that the coefficients of x and y in the second equation are twice the coefficients in the first equation. Specifically, 2x is 2 times x, and -6y is 2 times -3y. For the two equations to represent the same line, the constant term Q in the second equation must also be twice the constant term in the first equation. This is because if one equation is a multiple of the other, all corresponding terms must be in the same proportion. This concept is crucial for understanding how to manipulate equations to achieve desired solutions.

Therefore, we can write:

$$Q = 2 \times 4$$

This simple multiplication gives us the value of Q that makes the two equations equivalent. If the constant terms are in the same proportion as the coefficients of x and y, the equations will represent the same line, and the system will have infinitely many solutions. This is a fundamental principle in solving systems of linear equations.

Calculating this, we get:

$$Q = 8$$

This means that when Q is 8, the second equation 2x - 6y = Q becomes 2x - 6y = 8, which is exactly twice the first equation x - 3y = 4. Thus, the two equations are equivalent, and the system has infinitely many solutions. This result confirms our understanding of the conditions required for a system of equations to have infinitely many solutions.

To verify our result, let's substitute Q = 8 into the second equation:

$$2x - 6y = 8$$

Now, divide the entire equation by 2:

$$x - 3y = 4$$

This is the same as the first equation in the system. This confirms that when Q = 8, the two equations are identical, and the system has infinitely many solutions. The solution set is the set of all points (x, y) that satisfy the equation x - 3y = 4, as stated in the problem. This verification step is crucial to ensure the correctness of our solution and understanding of the underlying concepts.

Thus, we have successfully found the value of Q that makes the two equations represent the same line. This method of verification highlights the importance of checking solutions in mathematical problems to ensure accuracy.

In conclusion, the value of Q that makes the solution to the given system of equations the set (x, y) x - 3y = 4 is Q = 8. This occurs because the second equation becomes a multiple of the first equation, resulting in the two equations representing the same line. This leads to an infinite number of solutions, all lying on the line x - 3y = 4. The key to solving this problem was understanding the condition for infinite solutions in a system of linear equations: that the equations must be dependent and represent the same line. This article provides a comprehensive explanation of how to approach such problems, emphasizing the importance of understanding the underlying mathematical principles and verifying solutions. Systems of equations are a fundamental concept in mathematics, and mastering their solution techniques is essential for various applications in science and engineering. By understanding the conditions for different types of solutions, we can effectively solve a wide range of problems involving linear equations.