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. 3/8x - 1/2y = 7/8 -6x + 8y = -14
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Step 1: Write the system of equations in matrix form. The given system is: and . This can be written as an augmented matrix:
Step 2: Perform row operations to get a leading 1 in the first row, first column. Multiply the first row by to make the first element 1:
Step 3: Eliminate the first column of the second row by adding 6 times the first row to the second row:
Step 4: Interpret the resulting matrix. The second row indicates that the system has infinitely many solutions. The first row represents the equation .
Step 5: Express the solution in terms of . Solve for in terms of : . The solution set is .
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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 system's augmented matrix into reduced row echelon form, allowing for easy identification of solutions. This method can handle both unique solutions and cases with infinitely many solutions by manipulating the rows of the matrix through row operations.
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 a range of values. For example, in a two-variable system, if y is free, the solution can be written as x in terms of y, indicating that y can take any value.
An augmented matrix is a matrix that represents a system of linear equations, including the coefficients of the variables and the constants from the equations. It is formed by appending the constant terms as an additional column to the coefficient matrix. This format is essential for applying the Gauss-Jordan elimination method, as it simplifies the process of finding solutions to the system.