Use the Gauss-Jordan method to solve each system of equations. For systems in two variables with infinitely many solutions, write the solution with y arbitrary. For systems in three variables with infinitely many solutions, write the solution set with z arbitrary. See Examples 1-4. x + y = 5 x - y = -1
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Write the system of equations in augmented matrix form: .
Use row operations to get a leading 1 in the first row, first column (which is already done here).
Eliminate the first column of the second row by subtracting the first row from the second row: .
The new matrix becomes: .
Divide the second row by -2 to get a leading 1 in the second row, second column: .
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Key Concepts
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Gauss-Jordan Elimination
The Gauss-Jordan elimination method is a systematic procedure used to solve systems of linear equations. It involves transforming the augmented matrix of the system into reduced row echelon form (RREF) through a series of row operations. This method allows for easy identification of solutions, including unique solutions, no solutions, or infinitely many solutions.
A system of equations has infinitely many solutions when at least one equation can be derived from another, leading to dependent equations. In such cases, the solution can be expressed in terms of one or more free variables, allowing for multiple values that satisfy all equations. For example, in a two-variable system, one variable can be set as arbitrary, while the other is expressed in terms of it.
An augmented matrix is a matrix that represents a system of linear equations, combining the coefficients of the variables and the constants from the equations into a single matrix. This format simplifies the application of row operations during methods like Gauss-Jordan elimination, making it easier to manipulate and solve the system. The last column of the augmented matrix represents the constants from the equations.