When you solve a system of equations by the substitution method, how do you determine whether the system of equations is inconsistent?
The solution to a system of linear Eqs is the point where they intesect. If the lines are parallel, they cannot
intersect and, therefore, have no solution.
If we solve the Eqs of two parallel lines, the answer will not make sense.
For example, we could get -15 = 30 which is not true and doesn't make
sense.
We could easily calculate and compare the slopes before solving the Eqs.
If the slopes are equal, the lines are parallel and have no solution(inconsistent).
To determine whether a system of equations is inconsistent when using the substitution method, follow these steps:
1. Solve one of the equations for one variable in terms of the other variable. Let's call this variable 'x' for simplicity.
2. Substitute the expression for 'x' from step 1 into the other equation, replacing 'x' with the expression. This will give you a new equation with only one variable.
3. Solve the new equation for that variable.
4. If you obtain a solution for the variable in step 3, you have found a unique solution for the system of equations, and the system is consistent.
5. If solving the new equation results in a contradiction or an equation that cannot be satisfied, such as 2 = 0 or 0 = 0, then the original system of equations is inconsistent, meaning it has no solutions.
In summary, if you can find a solution for the variable in step 3, the system is consistent; otherwise, if you encounter a contradiction or unsatisfiable equation, the system is inconsistent.