Section 6.1 Systems of Linear Equations Chapter 6 Section 6.1 Systems of Linear Equations

Section 6.1 Systems of Linear 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)

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.

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.

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)

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:

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)

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

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

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

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:

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.

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: Corn Soybean Cottonseed Totals Fiber Fat Protein 10 30 20 40 25 1800 2800 2200

MORE EXAMPLES Example 5: Example 6: