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SECTION 6.1

- SYSTEMS OF LINEAR EQUATIONS
- SUBSTITUTION AND ELIMINATION

MOVIE THEATER TICKET SALES

SEE EXAMPLE 1 SEE EXAMPLE 2

EQUIVALENT SYSTEMS OF EQUATIONS

Linear System More than one linear equation

considered at a time. Solution - ordered pair (or

triple) that satisfies both (or all) equations

simultaneously.

CONSISTENT VS. INCONSISTENT

When a system of equations has at least one

solution, it is said to be consistent otherwise,

it is called inconsistent.

THREE POSSIBILITIES FOR A LINEAR SYSTEM

x - y 1 x - y 3

x - y 1 2x - y 4

x - y 1 2x - 2y 2

No Solution

One Solution

Infinitely Many Solutions

SOLVING A SYSTEM BY SUBSTITUTION

2x y 5 - 4x 6y 12

RULES FOR OBTAINING AN EQUIVALENT SYSTEM

1. Interchange any two equations. 2. Multiply (or

divide) each side of an equation by a nonzero

constant. 3. Replace any equation in the system

by the sum (or difference) of that equation and

a nonzero multiple of any other equation in the

system.

SOLVING A SYSTEM BY ELIMINATION

- 2x 3y 1
- x y - 3
- Multiply equation 2 by 2
- Replace equation 2 with the sum of equations 1

and 2.

MOVIE THEATER TICKET SALES

DO EXAMPLE 5

AN INCONSISTENT SYSTEM

2x y 5 4x 2y 8

AN DEPENDENT SYSTEM

2x y 4 - 6x - 3y - 12

3 EQUATIONS, 3 UNKNOWNS

2 x 4y - 2 z - 10 - 3x 4y - 2 z 5 5x

6y 3 z 3

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

5 5x 6y 3 z 3

Dividing 1st equation by 2 to make leading

coefficient equal to 1.

EXAMPLE

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

3x 6 y - 3 z - 15 - 3x 4y 2z 5

10y 5z -10

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

EXAMPLE

x 2 y - z - 5 5x 6y 3 z 3

-5x - 10 y 5 z 25 5x 6y 3z 3

- 4y 8z 28

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

EXAMPLE

Now we have 2 equations in only y z 10y -

5 z -10 - 4y 8 z 28

Divide 1st equation by 5 Divide 2nd equation by 4

EXAMPLE

2y - z - 2 - y 2 z 7

2y z -2 -2y 4z 14

3z 12 z 4

Multiply 2nd equation by 2 add to 1st

EXAMPLE

2y z -2 2y 4 -2 2y 2 y 1

x 2y z - 5 x 2 (1) - 4 - 5 x - 2 -

5 x - 3

(-3, 1, 4)

EXAMPLE

DO EXAMPLES 9, 10, 11

- CONCLUSION OF
- SECTION 6.1

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