the equation has at least one free variable
the homogeneous equation Ax = 0 has a nontrivial solution if and only if…
An equation having at least one free variable is an equation where the variable(s) is not completely restricted by any of the equations in the system. In simpler terms, there is at least one variable that can take a range of values without violating any of the constraints in the equation.
For example, in the equation 2x + 3y = 6, there is one free variable because we can choose any value for x, and then solve for y. However, we cannot freely choose any value for y because it must be related to x by the equation. In this case, the free variable is x, while y is dependent on the value chosen for x.
In contrast, in the equation 2x + 3y = 9, there are no free variables because any value of x will determine a unique value of y, and vice versa.
It is important to identify whether an equation has free variables or not, as this will determine whether it is possible to find a unique solution for the equation, or whether there is a range of solutions.
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