Section 4.1 Systems With Two Variables

OBJECTIVES The graphical method. Find solution of two linear equations using: A The graphical method.

OBJECTIVES The substitution method. Find solution of two linear equations using: B The substitution method.

OBJECTIVES The elimination method. Find solution of two linear equations using: C The elimination method.

OBJECTIVES Solve applications involving systems of equations. Find solution of two linear equations using: D Solve applications involving systems of equations.

Solving Two Equations in Two Unknowns by Elimination Clear any fractions or decimals.

Solving Two Equations in Two Unknowns by Elimination 2. Multiply both sides of the equations (as needed) by numbers that make the coefficients of one of the variables additive inverses.

Solving Two Equations in Two Unknowns by Elimination 3. Add the two equations. 4. Solve for the remaining variable.

Solving Two Equations in Two Unknowns by Elimination 5. Substitute this solution into one of the equations and solve for second variable. 6. Check the solution.

Exercise #2 Practice Test Chapter 4 Systems With Two Variables Section 4.1A Practice Test Exercise #2

Use the graphical method to solve the system.

Use the graphical method to solve the system.

Use the graphical method to solve the system. There is no solution. System is inconsistent. Lines are parallel.

Use the graphical method to solve the system. 5 x -5 5 y -5

Exercise #3 Practice Test Chapter 4 Systems With Two Variables Section 4.1A Practice Test Exercise #3

Use the graphical method to solve the system. x 5 -5

Use the graphical method to solve the system. Infinitely many solutions y x 5 -5

Exercise #5 Practice Test Chapter 4 Systems With Two Variables Section 4.1B Practice Test Exercise #5

Use the substitution method to solve the system.

Use the substitution method to solve the system. NO solution

Exercise #9 Practice Test Chapter 4 Systems With Two Variables Section 4.1C Practice Test Exercise #9

Solve the system. Multiply by 6. Multiply by 8. Multiply by –2.

Solve the system.

Solve the system. Substitute x = 4 in

Solve the system.

Solve the system. Solution is (4, 0).

Section 4.2 Systems with Three Variables

OBJECTIVES A Solve a system of three equations and three unknowns by the elimination method.

OBJECTIVES B Determine if a system of three equations in three unknowns is consistent, inconsistent, or dependent.

OBJECTIVES C Solve applications involving systems of three equations.

Three Equations in Three Unknowns by Elimination PROCEDURE FOR SOLVING Three Equations in Three Unknowns by Elimination Select a pair of equations and eliminate one variable from this pair.

Three Equations in Three Unknowns by Elimination PROCEDURE FOR SOLVING Three Equations in Three Unknowns by Elimination 2. Select a different pair of equations and eliminate the same variable as in step 1.

Three Equations in Three Unknowns by Elimination PROCEDURE FOR SOLVING Three Equations in Three Unknowns by Elimination 3. Solve the pair of equations resulting from step 1 and 2.

Three Equations in Three Unknowns by Elimination PROCEDURE FOR SOLVING Three Equations in Three Unknowns by Elimination 4. Substitute the values found in the simplest of original equations. Solve for third variable.

Three Equations in Three Unknowns by Elimination PROCEDURE FOR SOLVING Three Equations in Three Unknowns by Elimination 5. Check by substituting the values in each of the original equations.

Solving Three Equations in Three Unknowns by Elimination The system is consistent & independent; it has one solution consisting of an ordered triple (x, y, z).

Solving Three Equations in Three Unknowns by Elimination The system is inconsistent. It has no solution.

Solving Three Equations in Three Unknowns by Elimination The system is consistent and dependent. It has infinitely many solutions.

Exercise #11 Practice Test Chapter 4 Systems With Two Variables Section 4.2A Practice Test Exercise #11

Solve the system. x = 1

Solve the system. x = 1

Solve the system. x = 1

Section 4.3 Coin, Distance-Rate-Time, Investment and Geometry Problems

OBJECTIVES A Solve coin problems with two or more unknowns.

OBJECTIVES B Solve general problems with two or more unknowns.

OBJECTIVES C Solve rate, time and distance problems with two or more unknowns.

OBJECTIVES D Solve investment problems with two or more unknowns.

OBJECTIVES E Solve geometry problems with two or more unknowns.

Exercise #16 Practice Test Chapter 4 Systems With Two Variables Section 4.3C Practice Test Exercise #16

A motorboat can go 10 mi downstream on a river in 20 min A motorboat can go 10 mi downstream on a river in 20 min. It takes 30 min for this boat to go back upstream the same 10 mi. Find the speed of the current.

A motorboat can go 10 mi downstream on a river in 20 min A motorboat can go 10 mi downstream on a river in 20 min. It takes 30 min for this boat to go back upstream the same 10 mi. Find the speed of the current.

A motorboat can go 10 mi downstream on a river in 20 min A motorboat can go 10 mi downstream on a river in 20 min. It takes 30 min for this boat to go back upstream the same 10 mi. Find the speed of the current. (3) (4)

A motorboat can go 10 mi downstream on a river in 20 min A motorboat can go 10 mi downstream on a river in 20 min. It takes 30 min for this boat to go back upstream the same 10 mi. Find the speed of the current. The rate of the current is 5 mi/hr.

Section 4.4 Matrices

OBJECTIVES A Perform elementary operations on systems of equations.

OBJECTIVES B Solve systems of linear equations using matrices.

OBJECTIVES C Solve applications using matrices.

A rectangular array of numbers enclosed in brackets. DEFINITION Matrix A rectangular array of numbers enclosed in brackets.

Elementary Operations on Systems of Equations PROCEDURE Elementary Operations on Systems of Equations 1. The order of equations may be changed. This clearly cannot affect the solutions.

Elementary Operations on Systems of Equations PROCEDURE Elementary Operations on Systems of Equations 2. Any of the equations may be multiplied by any nonzero real number.

Elementary Operations on Systems of Equations PROCEDURE Elementary Operations on Systems of Equations 3. Any equation of a system may be replaced by the sum of itself and any other equation of the system.

Elementary Row Operations PROCEDURE Elementary Row Operations on Matrices Change the order of the rows.

Elementary Row Operations PROCEDURE Elementary Row Operations on Matrices 2. Multiply all elements of a row by any nonzero number.

Elementary Row Operations PROCEDURE Elementary Row Operations on Matrices 3. Replace any row by the element-by-element sum of itself and any other row.

Exercise #18 Practice Test Chapter 4 Systems With Two Variables Section 4.4A Practice Test Exercise #18

Use matrices to solve the system.

Use matrices to solve the system.

Use matrices to solve the system.

Use matrices to solve the system.

Use matrices to solve the system.

Use matrices to solve the system.

Use matrices to solve the system.

Section 4.5 Determinants and Cramer’s Rule

OBJECTIVES Evaluate a 2 by 2 determinant. A

OBJECTIVES B Use Cramer’s rule to solve a system of two equations in two unknowns.

OBJECTIVES C Use minors to evaluate 3 by 3 determinants.

OBJECTIVES D Use Cramer’s rule to solve a system of three equations.

Determinant

Cramer’s Rule - 2 Equations

Cramer’s Rule - 2 Equations

Cramer’s Rule - 2 Equations

Cramer’s Rule - 2 Equations

Cramer’s Rule - 2 Equations

Cramer’s Rule - 2 Equations

DEFINITION Minor The determinant that remains after deleting the row and column in which the element appears.

Minor

DEFINITION Sign Array

Cramer’s Rule - 3 Equations

Cramer’s Rule - 3 Equations

Cramer’s Rule - 3 Equations

Cramer’s Rule - 3 Equations

Cramer’s Rule - 3 Equations 2.

Cramer’s Rule - 3 Equations 3.

Exercise #19a Practice Test Chapter 4 Systems With Two Variables Section 4.5A Practice Test Exercise #19a

Evaluate.

Exercise #19b Practice Test Chapter 4 Systems With Two Variables Section 4.5A Practice Test Exercise #19b

Evaluate.

Exercise #20 Practice Test Chapter 4 Systems With Two Variables Section 4.5B Practice Test Exercise #20

Solve by Cramer’s rule.

Solve by Cramer’s rule.

Solve by Cramer’s rule.

Solve by Cramer’s rule.

Exercise #21 Practice Test Chapter 4 Systems With Two Variables Section 4.5C Practice Test Exercise #21

Evaluate.

Evaluate.

Exercise #23 Practice Test Chapter 4 Systems With Two Variables Section 4.5D Practice Test Exercise #23

Solve by Cramer’s rule.

Solve by Cramer’s rule.

Solve by Cramer’s rule.

Solve by Cramer’s rule.

Solve by Cramer’s rule.

Solve by Cramer’s rule.

Solve by Cramer’s rule.

Solve by Cramer’s rule.

Solve by Cramer’s rule.

Solve by Cramer’s rule.

Section 4.6 Systems of Linear Inequalities

OBJECTIVES A Graphing systems of two linear inequalities.

OBJECTIVES B Graphing systems of inequalities.

Graphing Inequalities PROCEDURE Graphing Inequalities

Graphing Inequalities PROCEDURE Graphing Inequalities Use a test point to shade the half-plane that is the graph of each linear inequality.

Graphing Inequalities PROCEDURE Graphing Inequalities Graph is the intersection of the half-planes, that is, the region consisting of the points satisfying all inequalities.

Exercise #25 Practice Test Chapter 4 Systems With Two Variables Section 4.6B Practice Test Exercise #25