Introduction: A homogeneous system of linear equations is a set of equations where the constant term in each equation is zero. In other words, it is a system of equations in which all the equations have the form Ax + By = 0. The goal is to determine whether there exists a non-zero solution for the system.

Consistency of a Homogeneous System: A homogeneous system of linear equations is always consistent. This means that there always exists at least one solution to the system. This can be proven by considering the trivial solution, where all the variables are set to zero.

Ratio X = 0: The statement mentions that if the ratio X = 0 is always a solution of the homogeneous system of equations, then we need to determine which statement is true.

Explanation: Let's consider a homogeneous system of equations with unknowns X and Y:

Equation 1: Ax + By = 0
Equation 2: Cx + Dy = 0
Equation 3: Ex + Fy = 0

If the ratio X = 0 is always a solution, it means that when X = 0, the equations become:

Equation 1: By = 0
Equation 2: Dy = 0
Equation 3: Fy = 0

Case 1: Non-Zero Y: If Y is non-zero, then the equations become inconsistent. This is because in order for the equations to be consistent, the coefficients of Y in all the equations should be the same. But in this case, the coefficients are different (B, D, and F). Therefore, Y must be zero for the system to be consistent.

Case 2: Y = 0: If Y = 0, then the equations become:

Equation 1: By = 0, which is always true regardless of the value of B.
Equation 2: Dy = 0, which is always true regardless of the value of D.
Equation 3: Fy = 0, which is always true regardless of the value of F.

In this case, the system is consistent because for any values of B, D, and F, the equations are satisfied when Y = 0.

Conclusion: Based on the above analysis, the statement that is true is "If the ratio X = 0 is always a solution of the homogeneous system of equations, then Y = 0." This means that for the system to be consistent, the value of Y must be zero.

Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

Any homogeneous linear equation(2 variable) will be of the form ax+by=0.

And (0,0) will always be solution of this.

So, Assertion is correct and reason is it's correct explanation.