Consistency of the System of Equations

Oct 13, 2025 by miodeya.com 35 views

Assertion (A): The value of ${$ p $}$ for which the system of equations ${$ 4x + py + 8 = 0 $}$ and ${$ 2x + 2y + 2 = 0 $}$ is consistent is 4.

Reason (R): The system of equations ${$ a_1x + b_1y = c_1 $}$ and ${$ a_2x + b_2y = c_2 $}$ is consistent if and only if the determinant of the coefficient matrix is zero.

Discussion

The given assertion is that the value of ${$ p $}$ for which the system of equations ${$ 4x + py + 8 = 0 $}$ and ${$ 2x + 2y + 2 = 0 $}$ is consistent is 4. To verify this, we need to find the value of ${$ p $}$ that makes the system of equations consistent.

System of Equations

The system of equations is given by:

${$ 4x + py + 8 = 0 $}$

${$ 2x + 2y + 2 = 0 $}$

We can rewrite the second equation as:

${$ x + y + 1 = 0 $}$

Coefficient Matrix

The coefficient matrix of the system of equations is given by:

${$ \begin{bmatrix} 4 & p \ 2 & 2 \end{bmatrix} $}$

The determinant of the coefficient matrix is given by:

${$ \begin{vmatrix} 4 & p \ 2 & 2 \end{vmatrix} = 4(2) - p(2) = 8 - 2p $}$

Consistency of the System of Equations

The system of equations is consistent if and only if the determinant of the coefficient matrix is zero. Therefore, we need to find the value of ${$ p $}$ that makes the determinant of the coefficient matrix zero.

${$ 8 - 2p = 0 $}$

Solving for ${$ p $}$, we get:

${$ p = 4 $}$

Therefore, the value of ${$ p $}$ for which the system of equations is consistent is indeed 4.

Verification of the Reason

The reason given is that the system of equations ${$ a_1x + b_1y = c_1 $}$ and ${$ a_2x + b_2y = c_2 $}$ is consistent if and only if the determinant of the coefficient matrix is zero. This is a well-known result in linear algebra, and it can be verified as follows:

Let ${$ A = \begin{bmatrix} a_1 & b_1 \ a_2 & b_2 \end{bmatrix} $}$ be the coefficient matrix of the system of equations. Then, the system of equations is consistent if and only if the equation ${$ Ax = c $}$ has a solution.

The equation ${$ Ax = c $}$ can be written as:

${$ \begin{bmatrix} a_1 & b_1 \ a_2 & b_2 \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix} = \begin{bmatrix} c_1 \ c_2 \end{bmatrix} $}$

This is equivalent to the system of equations:

${$ a_1x + b_1y = c_1 $}$

${$ a_2x + b_2y = c_2 $}$

The system of equations is consistent if and only if the determinant of the coefficient matrix is zero. This is because the determinant of the coefficient matrix is zero if and only if the two equations are linearly dependent, which means that the system of equations has a solution.

Therefore, the reason given is correct, and the value of ${$ p $}$ for which the system of equations is consistent is indeed 4.

Conclusion

In conclusion, the assertion that the value of ${$ p $}$ for which the system of equations ${$ 4x + py + 8 = 0 $}$ and ${$ 2x + 2y + 2 = 0 $}$ is consistent is 4 is correct. The reason given is also correct, and it can be verified using the properties of linear algebra. Therefore, the assertion is verified, and the value of ${$ p $}$ for which the system of equations is consistent is indeed 4.
Q&A: Assertion (A) and Reason (R)

Q: What is the value of ${$ p $}$ for which the system of equations ${$ 4x + py + 8 = 0 $}$ and ${$ 2x + 2y + 2 = 0 $}$ is consistent?

A: The value of ${$ p $}$ for which the system of equations is consistent is 4.

Q: Why is the system of equations consistent when ${$ p = 4 $}$?

A: The system of equations is consistent when ${$ p = 4 $}$ because the determinant of the coefficient matrix is zero. This means that the two equations are linearly dependent, and therefore, the system of equations has a solution.

Q: What is the determinant of the coefficient matrix?

A: The determinant of the coefficient matrix is given by:

${$ \begin{vmatrix} 4 & p \ 2 & 2 \end{vmatrix} = 4(2) - p(2) = 8 - 2p $}$

Q: How do we find the value of ${$ p $}$ that makes the determinant of the coefficient matrix zero?

A: We can find the value of ${$ p $}$ that makes the determinant of the coefficient matrix zero by setting the determinant equal to zero and solving for ${$ p $}$. This gives us:

${$ 8 - 2p = 0 $}$

Solving for ${$ p $}$, we get:

${$ p = 4 $}$

Q: What is the significance of the determinant of the coefficient matrix being zero?

A: The determinant of the coefficient matrix being zero means that the two equations are linearly dependent. This means that the system of equations has a solution, and therefore, the system of equations is consistent.

Q: Can we use the reason given to verify the assertion?

A: Yes, we can use the reason given to verify the assertion. The reason given is that the system of equations ${$ a_1x + b_1y = c_1 $}$ and ${$ a_2x + b_2y = c_2 $}$ is consistent if and only if the determinant of the coefficient matrix is zero. This is a well-known result in linear algebra, and it can be verified using the properties of linear algebra.

Q: What is the conclusion of the verification?

A: The conclusion of the verification is that the assertion is correct, and the value of ${$ p $}$ for which the system of equations is consistent is indeed 4.

Q: What is the significance of the verification?

A: The significance of the verification is that it provides a rigorous proof of the assertion. It shows that the assertion is not just a statement, but it is a mathematical fact that can be verified using the properties of linear algebra.

Q: Can we use the verification to solve other problems?

A: Yes, we can use the verification to solve other problems. The verification provides a general result that can be applied to other systems of equations. Therefore, it can be used to solve other problems that involve systems of equations.

Q: What is the final answer to the problem?

A: The final answer to the problem is that the value of ${$ p $}$ for which the system of equations ${$ 4x + py + 8 = 0 $}$ and ${$ 2x + 2y + 2 = 0 $}$ is consistent is 4.