Question: Assertion: Homogeneous system of linear equations is always consistent. Reason: \[x = 0,y = 0\] is...

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 incorrect but Reason is correct.

Answer 1

Solution

Here we will use the concept of the consistency. First, we will write the general form or general equation of the Homogeneous system of linear equations. Then we will check the equation consistency at point $\left( {0,0} \right)$ . If it satisfies the equation, then we will choose the option which best resembles the answer.

Complete step-by-step answer:
We know that the general equation of a Homogeneous system of linear equations with two variables in it is $ax + by = 0$ .
At the point $\left( {0,0} \right)$ , the equation is always consistent as this point is always the solution of the linear equation.
$\Rightarrow a\left( 0 \right) + b\left( 0 \right) = 0$
By solving the above equation we get
$\Rightarrow 0 = 0$ i.e. LHS is equal to the RHS of the equation.
So we can clearly see that $x = 0,y = 0$ is always a solution of the homogeneous system of equations with unknowns $x$ and $y$ .
Hence the Assertion is correct with the reason as its correct explanation.
So, option A is the correct option.

Note: Here the equation formed is the linear equation in which the highest exponent of the variable is one. Roots are those values of the equation where the value of the equation becomes zero. For any equation, numbers of roots are always equal to the value of the highest exponent of the variable. So, a linear equation will have only one solution. A quadratic equation is an equation in which the highest exponent of the variable is two and a quadratic equation has only two roots. Also, the cubic equations are the equation in which the highest exponent of the variable is 3.