If the third equation were x+3 y+3 z+4=0 and not as given, then for unique solution, (a) p=4 ; q

If the third equation were x+3 y+3 z+4=0 and not as given,...

Key Concepts

System of Linear Equations

A system of linear equations consists of multiple linear equations involving the same set of variables. The solution to such a system is the set of variable values that satisfy every equation simultaneously. Understanding the structure and coefficients in the system is critical for analyzing its properties, such as consistency and the number of solutions.

Unique Solution and Consistency

A unique solution in the context of a linear system means that exactly one set of values for the variables satisfies all the equations. For a system to have a unique solution, the equations must be independent, and their arrangement must avoid redundancy or contradictions. This often involves verifying that the system’s coefficient matrix is non-singular, ensuring the relationships between the equations are such that they intersect at a single point in the solution space.

Determinant and Non-singularity

The determinant of the coefficient matrix of a system of linear equations provides important information about the system's properties. A non-zero determinant indicates that the matrix is non-singular, which implies that the system of equations is independent and therefore has a unique solution. Conversely, a zero determinant suggests that the system may be either inconsistent or have infinitely many solutions.

Parameters in Linear Equations

Parameters in linear equations are constants that can modify the behavior or structure of the system when varied. They play a vital role in determining whether a system is consistent or has a unique solution. Adjusting parameter values can change the relationships among the equations, potentially leading to scenarios where a unique solution exists if and only if the parameters satisfy certain conditions.

Transcript

00:01 So in the given question we have a set of equations that are given as 2x plus 2x plus p y plus 6 z is equal to 8.

00:17 The second equation is x plus 2y plus x plus q is equal to 5 and x plus y plus x plus y plus qu z is equal to 5 and x plus y plus x plus is equal to 4.

00:33 So this is given in the question and now we are asked that if the third equation is replaced if the third equation is replaced with another equation that is given as x plus 3y plus 3 z is equal to minus 4 then what would be the condition for this system of equations to have a unique solution have a unique solution.

01:08 So the condition for a system of equations to have a unique solution is that when we take the determinant that is formed by the coefficients in the system of equations we would get the value of the determinant as a value which is not equal to 0 right so determinant not equal to 0 is so determinant not the condition for the system to have a unique solution.

01:38 So let's find the determinant of the system of equations in which we are taking the coefficients of the system or the equation right the coefficients in the system of equations are 2 p6 1 2 cube and 1 3 and when we find the determinant what we would have is 2 times 6 minus 3 cube then we have minus p times 3 minus q and we have 6 times 3 minus 2 which is equal to 1 and the value of this determinant should not be equal to 0 so on expanding this this value we would have 12 minus 6 q minus 3p plus p q plus 6 is equal to is not equal to 0 12 plus 6 is equal to 18 so we can write minus 6 q plus 18 minus 3p plus p q should not be equal to 0 and from this from this two terms let's take minus 6 q plus 18 as a common factor then we would have q minus 3 minus 6 times q minus 3 and from these two terms let's take p as a common factor right so plus p times q minus 3 is not equal to 0 and what we can write from this this is that we can take q minus 3 as a common factor now and we would have q minus 3 times p minus 6 is not equal to 0...

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