In Exercises 1 - 24, use Gaussian Eliminaion to find the complete solution to each system of equations, or show that none exists.
8x + 5y + 11z = 30
- x - 4y + 2z = 3
2x - y + 5z = 12
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1
Write the system of equations as an augmented matrix:
Use row operations to get a leading 1 in the first row, first column. You can start by swapping rows if necessary or multiplying/dividing a row by a constant.
Eliminate the first column entries below the leading 1 by adding suitable multiples of the first row to the other rows.
Move to the second row, second column, and get a leading 1 by using row operations. Then eliminate the entries above and below this leading 1.
Continue to the third row, third column, and get a leading 1. Use row operations to eliminate the entries above this leading 1, achieving row-echelon form.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Gaussian Elimination
Gaussian elimination is a method for solving systems of linear equations. It involves transforming the system's augmented matrix into row echelon form using a series of row operations, which simplifies the equations. Once in this form, back substitution can be used to find the values of the variables. This technique is fundamental for determining whether a unique solution, infinitely many solutions, or no solution exists.
Row operations are the basic manipulations applied to the rows of a matrix during Gaussian elimination. These include swapping two rows, multiplying a row by a non-zero scalar, and adding or subtracting a multiple of one row to another. Understanding these operations is crucial, as they maintain the equivalence of the system while simplifying it, allowing for easier identification of solutions.
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 representation is essential for applying Gaussian elimination, as it allows for a systematic approach to solving the equations and analyzing the solution set.