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Systems of Linear Equations: Solution Methods and Gauss-Jordan Method

Learn about solving systems of linear equations using substitution, elimination, and the Gauss-Jordan method. Understand the concepts of dependent and inconsistent systems. Examples and word problems provided.

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Systems of Linear Equations: Solution Methods and Gauss-Jordan Method

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  1. Chapter 6 Section 6.1 Systems of Linear Equations

  2. Section 6.1Systems of Linear Equations • Equations • Linear equations (1st degree equations) • System of linear equations • Solution of the system • System of 2 linear equations in 2 variables: independent, dependent, inconsistent systems • Solution methods: substitution and elimination. • Example 1 (p. 297)

  3. Larger system of linear equations Two systems are equivalent if they have the same solutions. Elementary operations (to produce an equivalent system): • Interchange any two equations • Multiply both sides of an equation by a non-zero constant. • Replace an equation by the sum of itself and a constant multiple of another equation in the system.

  4. Elimination method Example 6 (p. 300) Elimination method for solving larger system of linear equations: • Make the leading coefficient of the first equation 1. • Eliminate the leading variable of the first equation from each later equation. • Repeat steps 1 and 2 for the second equation. • Repeat steps 1 and 2 for the third, fourth equation and so on, till the last equation. • Then solve the resulting system by back substitution.

  5. MATRIX METHODS • Matrix • Row, Column, Element (entry) • Augmented matrix Row operations on matrices: • Interchange any two rows. • Multiply each element of a row by a non-zero constant. • Replace a row by the sum of itself and a constant multiple of another row of the matrix. Example 7 (p. 303)

  6. MATRIX METHODS Row echelon form: • All rows having entirely zeros (if any) are at the bottom • The first nonzero entry in each row is 1 (called leading 1). • Each leading 1 appears to the right of the leading 1’s in any preceding rows. Example:

  7. DEPENDENT AND INCONSISTENT SYSTEMS • Example 9: • Solution: The system has infinitely many solutions (the system is dependent) • Example 11: • Solution: the system has no solution (it is inconsistent)

  8. 6.2 GAUSS-JORDAN METHOD Example 1: (The system is independent)

  9. 6.2 GAUSS-JORDAN METHOD Example 2: (The system is inconsistent)

  10. 6.2 GAUSS-JORDAN METHOD Example 3: (The system is dependent)

  11. GAUSS-JORDAN METHOD A matrix is said to be in reduced row echelon form if it is in row echelon form and every column containing a leading 1 has zeros in all its other entries. Example:

  12. GAUSS-JORDAN METHOD • Arrange the equations with the variables terms in the same order on the left of the equal sign and the constants on the right. • Write the augmented matrix of the system. • Use the row operations to transform the augmented matrix into reduced row echelon form: • Stop the process in step 3 if you obtain a row whose elements are all zeros except the last one. In that case, the system is inconsistent and has no solutions. Otherwise, finish step 3 and read the solutions of the system from the final matrix.

  13. Example 5: Word Problem An animal feed is to be made from corn, soybean, and cottonseed. Determine how many units of each ingredient are needed to make a feed that supplies 1800 units of fiber, 2800 units of fat, and 2200 units of protein, given the information below:

  14. MORE EXAMPLES • Example 5: • Example 6:

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