Section 4 1 Systems With Two Variables OBJECTIV
Section 4. 1 Systems With Two Variables
OBJECTIV Find solution of two ES linear equations using: A The graphical method.
OBJECTIV Find solution of two ES linear equations using: B The substitution method.
OBJECTIV Find solution of two ES linear equations using: C The elimination method.
OBJECTIV Find solution of two ES linear equations using: D Solve applications involving systems of equations.
Solving Two Equations in Two Unknowns by Elimination 1. Clear any fractions or 2. 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 inverse
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 in one of the equations and solve for second variable 6. Check the solution.
Chapter 4 Systems With Two Variables Section 4. 1 A 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
Chapter 4 Systems With Two Variables Section 4. 1 A Practice Test Exercise #3
Use the graphical method to solve the system. y x 5 -5
Use the graphical method to solve the system. Infinitely many solutions y x 5 -5
Chapter 4 Systems With Two Variables Section 4. 1 B Practice Test Exercise #5
Use the substitution method to solve the system.
Use the substitution method to solve the system. NO solution
Chapter 4 Systems With Two Variables Section 4. 1 C 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
OBJECTIV ES A Solve a system of three equations and three unknowns by the elimination method.
OBJECTIV ES B Determine if a system of three equations in three unknowns is consistent, inconsistent, or
OBJECTIV ES C Solve applications involving systems of three equations.
PROCEDURE FOR SOLVING Three Equations in Three Unknowns by Elimination 1. Select a pair of equations 2. and eliminate one variab 3. from this pair.
PROCEDURE FOR SOLVING Three Equations in Three Unknowns by Elimination 2. Select a different pair of equations and eliminate th same variable as in step 1
PROCEDURE FOR SOLVING Three Equations in Three Unknowns by Elimination 3. Solve the pair of equation resulting from step 1 and
PROCEDURE FOR SOLVING Three Equations in Three Unknowns by Elimination 4. Substitute the values foun in the simplest of original equations. Solve for third variable.
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 1. The system is consistent & independent; it has one solution consisting of
Solving Three Equations in Three Unknowns by Elimination 2. The system is inconsistent. 3. It has no solution.
Solving Three Equations in Three Unknowns by Elimination 3. The system is consistent and dependent. It has infinitely many solutions.
Chapter 4 Systems With Two Variables Section 4. 2 A 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
OBJECTIV ES A Solve coin problems with two or more unknowns.
OBJECTIV ES B Solve general problems with two or more unknowns.
OBJECTIV ES C Solve rate, time and distance problems with two or more unknowns.
OBJECTIV ES D Solve investment problems with two or more unknowns.
OBJECTIV ES E Solve geometry problems with two or more unknowns.
Chapter 4 Systems With Two Variables Section 4. 3 C Practice Test Exercise #16
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. 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. 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. 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
OBJECTIV ES A Perform elementary operations on systems of equations.
OBJECTIV ES B Solve systems of linear equations using matrices.
OBJECTIV ES C Solve applications using matrices.
DEFINITION Matrix A rectangular array of numbers enclosed in brackets.
PROCEDURE Elementary Operations on Systems of Equations 1. The order of equations may be changed. This clearly cannot affect the solutions.
PROCEDURE Elementary Operations on Systems of Equations 2. Any of the equations may be multiplied by any nonzero real number.
PROCEDURE Elementary Operations on Systems of Equations 3. Any equation of a system may be replaced by the su of itself and any other equation of the system.
PROCEDURE Elementary Row Operations on Matrices 1. Change the order of the 2. rows.
PROCEDURE Elementary Row Operations on Matrices 2. Multiply all elements of a row by any nonzero numb
PROCEDURE Elementary Row Operations on Matrices 3. Replace any row by the element-by-element sum o itself and any other row.
Chapter 4 Systems With Two Variables Section 4. 4 A 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
OBJECTIV ES A Evaluate a 2 by 2 determinant.
OBJECTIV ES B Use Cramer’s rule to solve a system of two equations in two unknowns.
OBJECTIV ES C Use minors to evaluate 3 by 3 determinants.
OBJECTIV ES 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 remain after deleting the row and column in which the elemen 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.
Chapter 4 Systems With Two Variables Section 4. 5 A Practice Test Exercise #19 a
Evaluate.
Chapter 4 Systems With Two Variables Section 4. 5 A Practice Test Exercise #19 b
Evaluate.
Chapter 4 Systems With Two Variables Section 4. 5 B Practice Test Exercise #20
Solve by Cramer’s rule.
Solve by Cramer’s rule.
Solve by Cramer’s rule.
Solve by Cramer’s rule.
Chapter 4 Systems With Two Variables Section 4. 5 C Practice Test Exercise #21
Evaluate.
Evaluate.
Chapter 4 Systems With Two Variables Section 4. 5 D 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
OBJECTIV ES A Graphing systems of two linear inequalities.
OBJECTIV ES B Graphing systems of inequalities.
PROCEDURE Graphing Inequalities
PROCEDURE Graphing Inequalities 2. Use a test point to shade 3. the half-plane that is the 4. graph of each linear 5. inequality.
PROCEDURE Graphing Inequalities 3. Graph is the intersection of 4. the half-planes, that is, the 5. region consisting of
Chapter 4 Systems With Two Variables Section 4. 6 B Practice Test Exercise #25
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