The system will have infinite number of solutions if (a) p=2 (b) q=3 (c) p ≠2 ; q ≠3 (d) p ≠2 ;

The system will have infinite number of solutions if (a) p=2...

Key Concepts

Rank and Free Variables

The rank of the coefficient matrix of a system reflects the number of linearly independent equations. If the rank is less than the number of variables, there will be free variable(s) in the system, which is exactly the condition that gives rise to infinitely many solutions. Determining rank in the presence of parameters is crucial for establishing when the system becomes underdetermined.

Parameter Conditions in Linear Systems

When a system of equations includes parameters, certain combinations of these parameters can lead to special cases like dependent equations or changes in the rank of the system. Understanding and identifying these conditions is essential to determine for which values of the parameters the system will have an infinite number of solutions.

Consistency of Linear Systems

A system of linear equations is consistent if it has at least one solution. When parameters are involved, certain values may cause the system to be redundant, making the equations dependent and leading to infinitely many solutions. This key concept involves checking if the equations do not contradict each other when the parameters take specific values.

Dependent Equations

Dependent equations occur when one equation in a system can be expressed as a scalar multiple or linear combination of another. In the context of parameterized systems, particular parameter values can make the equations dependent, reducing the number of distinct equations and resulting in an infinite number of solutions.

Transcript

00:01 So in the given question we have a system of equations that are given us 2x plus p y plus 6 z plus 6 z is equal to 8 and we have x plus 2y plus qz is equal to 5 and the third equation is x plus y plus 3 z is equal to is equal to 4 and in the question we are asked that what should be the values of p and q for this system of equations to have to have infinite number of solutions infinite number of solutions so for a system to have a system to have infinite number of solutions what we should know is that we should calculate we should know that if a system has infinite number of solutions then the determinants delta 1 delta 2 and delta 3 should be equal to 0 what are the determinants delta 1 delta 2 and delta 3 right so delta 1 delta is the determinant which is formed by the coefficients of the matrix of the the coefficients of this the equations in the system of equations right it is a matrix formed by it is the determinant of a matrix formed by the coefficients in the system of equations so we have the determinant delta as 2p 6 x sorry y 1 2 q and 1 1 3 so we have listed the coefficients in this system of in this system of equations and the matrix so formed is called the coefficient matrix and delta is the determinant of this coefficient matrix right so now that we know delta what is delta one right so delta one is the matrix that is formed by replacing the first column of the matrix delta with the constants in in the question right so the constants that are given in the question are 8 5 and 4th right so when we replace the first column with these constants we call this determinant delta 1 right so if delta 1 is equal to 0 let's just calculate the determinant of this is delta 1 so let's write delta 2 and delta 3 as well, right? so delta 3 would then be 211 -8 -5 -4 6 q3.

03:16 So we replaced the second column with the constants in the equation and in the same way we can replace the third column next to write delta 3, right? so now that we have delta 1 delta 20.

03:35 In the question in the question there are options which gives us the answers of different values of p and q for which the system has infinite solutions so in all the options we can see that the answers are in between the numbers 2 and 3 right it is given that in some options it is given that for p equal to 2 the system has infinite number of solutions in another option it says for q equal to 3 the system has infinite number of solutions and so on.

04:11 So let's check these answers to see if the options given in the question are correct, right? so first what we are going to do is we are going to take p equal to 2.

04:24 So if we take p equal to 2, delta 1 would be 8 to 6, 5 to 1, 5 2 cube, 5 2 cube and 413...

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