To mentally check whether a linear system of two equations in two unknowns is independent and consistent, we can follow these steps:
1) Count the number of unknowns (variables) and the number of equations given in the system. In this case, there are two unknowns and two equations.
2) Compare the number of unknowns and the number of equations. If the number of unknowns is equal to the number of equations (both are 2 in this case), it suggests that the system has a chance of being independent and consistent.
3) Evaluate the determinant of the coefficient matrix (the matrix formed by the coefficients of the variables in the system).
- If the determinant is non-zero, it implies that the system is independent and consistent, meaning there is a unique solution.
- If the determinant is zero, it implies that the system can either be dependent (having infinitely many solutions) or inconsistent (having no solution).
The order in which we perform these checks would be:
1) Compare the number of unknowns and equations.
2) Evaluate the determinant of the coefficient matrix.
We perform these checks in this order because comparing the number of unknowns and equations gives us an immediate indication of whether the system has a chance of being independent and consistent. If the numbers don't match up, we can conclude that the system is inconsistent. However, if the numbers do match up, we proceed to check the determinant to confirm if the system is indeed independent and consistent or not. Evaluating the determinant helps us further narrow down the possibilities of the system being dependent or inconsistent.
You are given a linear system of two equations in two unknowns. Before solving, describe how you can mentally check whether the system is independent and consistent. In which order would you do your check? Why?
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