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?
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.
To mentally check whether a linear system of two equations in two unknowns is independent and consistent, you can follow these steps in the given order:
1. Check the number of solutions: The first step is to determine whether the system has a unique solution, infinitely many solutions, or no solutions. This can be done by comparing the coefficients of the variables' terms and examining any constants involved.
2. Determine the nature of the solutions: If the system has unique solutions, then it is independent and consistent. However, if the system has infinitely many solutions, it is dependent and consistent. If there are no solutions, then the system is inconsistent.
The reason for this order is that step 1 allows you to quickly identify if the system has any solutions at all. This helps to eliminate the need to perform further analysis if the system is inconsistent. Once you confirm the existence of solutions, step 2 allows you to determine if the system is dependent or independent based on the number of solutions.
Overall, by following this order, you can mentally check the independence and consistency of a linear system of two equations efficiently and effectively.
To mentally check whether the given linear system of two equations in two unknowns is independent and consistent, you can follow the steps below:
1. Check the coefficient matrix's determinant: The determinant of the coefficient matrix can be computed mentally. If the determinant is non-zero, then the system is independent and consistent. If the determinant is zero, proceed to the next step.
2. Compare the slopes of the equations: If the determinant is zero, it indicates that the two equations are either parallel or coincident. Mentally visualize the equations and compare their slopes:
a. If the slopes of the equations are different, the system is inconsistent, and there is no solution.
b. If the slopes of the equations are the same, proceed to the next step.
3. Compare the constant terms of the equations: If the slopes of the equations are the same, mentally compare the constant terms (also known as the y-intercepts) of the equations:
a. If the constant terms are also the same, the system is consistent and has infinitely many solutions.
b. If the constant terms are different, the system is inconsistent, and there is no solution.
The order of the check is important because calculating the determinant only requires mental arithmetic, making it a quick and efficient first step. If the determinant is non-zero, we can conclude that the system is independent and consistent without performing any further checks. However, if the determinant is zero, we need to compare the slopes to determine whether the system is inconsistent or has a unique solution. Finally, comparing the constant terms helps us identify whether the system is consistent with infinitely many solutions or inconsistent.