When you solve a system of equations by the substitution method, how do you determine whether the system of equations is dependent?

If your variables disappear as you simplify, and you end up with
0=0 (or k=k) the system is dependent.
(really just the same line, one equation is a multiple of the other)

If your variables disappear as you simplify, and you end up with
0=k (or c=k) the system is inconsistent
(Really just two parallel lines)

To determine whether a system of equations is dependent when solving by the substitution method, you should look for two specific cases:

1. If your variables disappear as you simplify, and you end up with 0=0 or k=k, then the system is dependent. This means that the two equations represent the same line or that one equation is a multiple of the other.

2. If your variables disappear as you simplify, and you end up with 0=k or c=k, then the system is inconsistent. This means that the two equations represent parallel lines that do not intersect.

By identifying these scenarios while simplifying the equations during the substitution method, you can determine whether the system of equations is dependent or inconsistent.

To determine whether a system of equations is dependent or inconsistent when solving it by the substitution method, you need to observe what happens when you simplify the equations.

If, after simplifying, the variables completely disappear and you end up with a true statement like 0 = 0 or k = k (where k can be any constant), then the system is dependent. This means that the two equations represent the same line, just written in different forms or with different coefficients. In other words, one equation is a multiple of the other. When the equations are dependent, you have infinitely many solutions because any value for the variables that satisfies one equation will also satisfy the other equation.

If, after simplifying, the variables completely disappear but you end up with a false statement like 0 = k or c = k (where k can be any constant), then the system is inconsistent. This means that the two equations represent parallel lines that will never intersect. When the equations are inconsistent, there are no solutions that satisfy both equations simultaneously.

So, by observing whether the variables disappear and the resulting statement being either true or false, you can determine whether a system of equations is dependent or inconsistent when solving it using the substitution method.