SECTION 6.1SECTION 6.1

SYSTEMS OF LINEAR SYSTEMS OF LINEAR EQUATIONS: EQUATIONS:

SUBSTITUTION AND SUBSTITUTION AND ELIMINATIONELIMINATION

MOVIE THEATER TICKET SALES

MOVIE THEATER TICKET SALES

SEE EXAMPLE 1SEE EXAMPLE 1

SEE EXAMPLE 2SEE EXAMPLE 2

EQUIVALENT SYSTEMS OF EQUATIONS

EQUIVALENT SYSTEMS OF EQUATIONS

Linear System Linear System

More than one linear equation More than one linear equation considered at a time.considered at a time.

Solution - ordered pair (or Solution - ordered pair (or triple) that satisfies both (or triple) that satisfies both (or all) equations simultaneously.all) equations simultaneously.

CONSISTENT VS. INCONSISTENT

CONSISTENT VS. INCONSISTENT

When a system of equations When a system of equations has at least one solution, it is has at least one solution, it is said to be consistent; said to be consistent; otherwise, it is called otherwise, it is called inconsistent.inconsistent.

THREE POSSIBILITIES FOR A LINEAR

SYSTEM

THREE POSSIBILITIES FOR A LINEAR

SYSTEMx - y = x - y = 11

x - y = x - y = 33 No No SolutionSolution

x - y = x - y = 11

2x - y = 2x - y = 44One One

SolutionSolution

x - y = 1x - y = 1

2x - 2y = 2x - 2y = 22InfinitelInfinitely Many y Many SolutionSolution

ss

SOLVING A SYSTEM BY SUBSTITUTION

SOLVING A SYSTEM BY SUBSTITUTION

2x + y = 52x + y = 5

- 4x + 6y = - 4x + 6y = 1212

RULES FOR OBTAINING AN EQUIVALENT

SYSTEM

RULES FOR OBTAINING AN EQUIVALENT

SYSTEM

1.1. Interchange any two equations.Interchange any two equations.

2.2. Multiply (or divide) each side of Multiply (or divide) each side of an an equation by a nonzero equation by a nonzero

constant.constant.

3.3. Replace any equation in the Replace any equation in the system by system by the sum (or difference) of the sum (or difference) of that equation that equation and a nonzero and a nonzero multiple of any other multiple of any other equation in equation in the system.the system.

SOLVING A SYSTEM BY ELIMINATION

SOLVING A SYSTEM BY ELIMINATION

2x + 3y = 12x + 3y = 1

-x + y = - 3x + y = - 3

Multiply equation 2 by 2Multiply equation 2 by 2

Replace equation 2 with the sum of Replace equation 2 with the sum of equations 1 and 2.equations 1 and 2.

MOVIE THEATER TICKET SALES

MOVIE THEATER TICKET SALES

DO EXAMPLE 5DO EXAMPLE 5

AN INCONSISTENT SYSTEM

AN INCONSISTENT SYSTEM

2x + y = 52x + y = 5

4x + 2y = 84x + 2y = 8

AN DEPENDENT SYSTEM

AN DEPENDENT SYSTEM

2x + y = 42x + y = 4

- 6x - 3y = - 12- 6x - 3y = - 12

3 EQUATIONS, 3 UNKNOWNS

3 EQUATIONS, 3 UNKNOWNS

2 x + 4y - 2 z = 2 x + 4y - 2 z = - 10- 10

- 3x + 4y - 2 z = - 3x + 4y - 2 z = 5 5

5x + 6y + 3 z = 5x + 6y + 3 z = 3 3Dividing 1st equation by 2 to Dividing 1st equation by 2 to make leading coefficient make leading coefficient equal to 1.equal to 1.

x + 2 y - z = x + 2 y - z = - 5 - 5

- 3x + 4y - 2 z = - 3x + 4y - 2 z = 5 5

5x + 6y + 3 z = 5x + 6y + 3 z = 3 3

EXAMPLEEXAMPLE

x + 2 y - z = x + 2 y - z = - 5 - 5

- 3x + 4y - 2 z = - 3x + 4y - 2 z = 5 5

Mult. 1st eqn by 3 and add to Mult. 1st eqn by 3 and add to 2nd2nd

3x + 6 y - 3 z = 3x + 6 y - 3 z = - 15- 15

- 3x + 4y – 2z = - 3x + 4y – 2z = 55

10y – 5z = 10y – 5z = -10-10

EXAMPLEEXAMPLE

x + 2 y - z x + 2 y - z = - 5= - 5

5x + 6y + 3 z 5x + 6y + 3 z = 3= 3

Mult. 1st eqn by -5 and add to Mult. 1st eqn by -5 and add to 3rd3rd

-5x - 10 y + 5 z -5x - 10 y + 5 z = 25= 25

5x + 6y + 3z = 5x + 6y + 3z = 3 3

- 4y + 8z = - 4y + 8z = 2828

EXAMPLEEXAMPLE

Now we have 2 equations in only Now we have 2 equations in only y & z: y & z:

10y - 5 z = -1010y - 5 z = -10

- 4y + 8 z = 28- 4y + 8 z = 28

Divide 1Divide 1stst equation by 5 equation by 5

Divide 2nd equation by 4Divide 2nd equation by 4

EXAMPLEEXAMPLE

2y - z = - 22y - z = - 2

- y + 2 z = 7- y + 2 z = 7

Multiply 2Multiply 2ndnd equation by 2 & equation by 2 & add to 1stadd to 1st

2y – z = -2 2y – z = -2

-2y + 4z = 14 -2y + 4z = 14

3z = 3z = 1212

z = z = 44

EXAMPLEEXAMPLE

2y – z = -22y – z = -2

2y – 4 = -22y – 4 = -2

2y = 22y = 2

y = 1y = 1

x + 2y – z = - 5x + 2y – z = - 5

x + 2 (1) - 4 = x + 2 (1) - 4 = - 5- 5

x - 2 = - 5x - 2 = - 5

x = - 3x = - 3

(-3, 1, 4)(-3, 1, 4)

EXAMPLEEXAMPLE

DO EXAMPLES 9, 10, 11DO EXAMPLES 9, 10, 11

CONCLUSION OF CONCLUSION OF SECTION 6.1SECTION 6.1