Question

Assertion: Homogeneous system of linear equations is always consistent.
Reason: \[x = 0,y = 0\] is always a solution of the homogeneous system of equations with unknowns x and \[y\].
Then which of the following statements is true?
A.Both Assertion and Reason are correct and Reason is the correct explanation for Assertion.
B.Both Assertion and Reason are correct and Reason is not the correct explanation for Assertion.
C.Assertion is correct but reason is incorrect.
D.Assertion is correct but reason is incorrect.

Answer 1

Hint: Here, we will use the concept of consistency. We will first express the general form of a homogeneous system of linear equations with two variables. Then, we will check if \[x = 0,y = 0\] is a solution to the linear equations or not.

Complete step-by-step answer:
A linear equation is said to be homogeneous if its constant part is zero. A homogeneous system of linear equations is a system in which all the linear equations are homogeneous. In the given problem, we are dealing with two unknown variables. So, a homogeneous system of linear equations with two unknown variables can have at most two equations.
Let us consider the following system:
\[ax + by = 0\] ………………\[\left( 1 \right)\]
\[cx + dy = 0\]……………...\[\left( 2 \right)\]
Here, \[a,b,c,d\] are constants which cannot all be simultaneously zero.
A homogeneous system of linear equations is said to be consistent if it possesses at least one solution.
Now, we will check if \[x = 0,y = 0\] is a solution to these equations.
We will first consider the equation \[\left( 1 \right)\].
LHS \[ = ax + by\]
Substituting \[x = 0,y = 0\] in the above expression, we get
\[ \Rightarrow \] LHS \[ = a(0) + b(0) = 0 = \] RHS
Now, we will consider the equation \[\left( 2 \right)\].
LHS \[ = cx + dy\]
Substituting \[x = 0,y = 0\] in the above expression, we get
\[ \Rightarrow \] LHS \[ = c(0) + d(0) = 0 = \] RHS
Hence, \[x = 0,y = 0\] is a solution to the homogeneous system of linear equations and so, it is consistent.
Therefore, both Assertion and Reason are correct and Reason is the correct explanation for Assertion.
Thus, the right option is A.

Note: A linear equation is an equation in which the highest degree of the variable is 1. A homogeneous system of linear equations is always consistent, in the sense that \[x = 0,y = 0\] is always a solution to the system. This solution is called a trivial solution. Any solution, where \[x \ne 0,y \ne 0\], then it is called a non-trivial solution.