A consistent system of linear equations is a set of equations that has at least one solution that satisfies all the equations simultaneously. In other words, the equations in the system are compatible and have a common solution that makes all the equations true.
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A consistent system of linear equations can have either a unique solution or an infinite number of solutions, depending on the number of equations and variables in the system.
The rank of the augmented matrix of a consistent system must be equal to the rank of the coefficient matrix, which ensures the system has at least one solution.
Graphically, a consistent system of two linear equations in two variables is represented by two intersecting lines, with the point of intersection being the unique solution.
Gaussian elimination and row reduction can be used to determine whether a system of linear equations is consistent and to find the solution(s) if the system is consistent.
The method of substitution or the method of elimination can also be used to solve a consistent system of linear equations with two variables.
Review Questions
Explain the relationship between the rank of the augmented matrix and the rank of the coefficient matrix for a consistent system of linear equations.
For a consistent system of linear equations, the rank of the augmented matrix must be equal to the rank of the coefficient matrix. This ensures that the system has at least one solution that satisfies all the equations simultaneously. If the ranks are not equal, the system would be inconsistent and have no solution that makes all the equations true.
Describe the graphical representation of a consistent system of two linear equations in two variables.
When graphed, a consistent system of two linear equations in two variables is represented by two intersecting lines. The point of intersection between the two lines is the unique solution that satisfies both equations. If the lines are parallel, the system is inconsistent and has no solution. If the lines are the same, the system is dependent and has an infinite number of solutions.
Analyze the steps involved in using Gaussian elimination or row reduction to determine the consistency and find the solution(s) of a system of linear equations.
To determine the consistency of a system of linear equations using Gaussian elimination or row reduction, the key step is to examine the rank of the augmented matrix and the rank of the coefficient matrix. If the ranks are equal, the system is consistent and has at least one solution. The solution(s) can then be found by performing row reduction on the augmented matrix to obtain the reduced row echelon form, which will reveal the values of the variables that satisfy all the equations in the system.
An inconsistent system of linear equations is a set of equations that has no solution that satisfies all the equations simultaneously. The equations in the system are incompatible and have no common solution.
A dependent system of linear equations is a set of equations where one equation is a scalar multiple of another equation in the system. This means the equations are not independent and have an infinite number of solutions.
An augmented matrix is a matrix that represents a system of linear equations, where the coefficients of the variables are in the left part of the matrix, and the constants on the right side of the equations are in the right part of the matrix.