SOLVING SYSTEMS OF LINEAR EQUATIONS An equation is said to be linear if every variable has degree...

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SOLVING SYSTEMS OF LINEAR EQUATIONS

• An equation is said to be linear if every variable has degree equal to one (or zero)

• is a linear equation

• is NOT a linear equation

8254 zyx

832 13 zyx

Review these familiar techniques for solving 2 equations in 2 variables. The same techniques will be

extended to accommodate larger systems.

Times 3

Substitute to solve: y=1

L1 represents line one L2 represents line two

L1 is replaced by 3L1

L1 is replaced by L1 + L2

L1 is replaced by (1/7)L1

These systems are said to be EQUIVALENT because they have the SAME SOLUTION.

PERFORM ANY OF THESE OPERATIONS ON A SYSTEM OF LINEAR EQUATIONS TO

PRODUCE AN EQUIVALENT SYSTEM:

• INTERCHANGE two equations (or lines)

• REPLACE Ln with k Ln , k is NOT ZERO

• REPLACE Ln with Ln + cLm

• note: Ln is always part of what replaces it.

EXAMPLES:

is equivalent to

L3 + 2L1

343 zyx

042 zyxReplace L1 with (1/2) L1

Replace L3 with L3 + L2

042 zyx

6x +4zReplace L1 withL1 + 2 L2

zxReplace L1 with L1 + - 4 L3

is EQUIVALENT to

To solve the following system, we look for an equivalent systemwhose solution is more obvious. In the process, we manipulateonly the numerical coefficients, and it is not necessary to rewritevariable symbols and equal signs:

This rectangular arrangement of numbers is called a MATRIX

Replace L1 with L1 + 2 L2

Replace L2 with L2 + 1L1

Interchange L2 and L3

Replace L1 with L1 + -1 L2

Replace L3 with -1 L3

Replace L2 with -1 L2

The original matrix representsa system that is equivalent to this final matrix whose solution is obvious

1001 x

2100 z

1010 y

The original matrix representsa system that is equivalent to this final matrix whose solution is obvious

The zeros

The diagonal of ones

1001 x

2100 z

1010 y

Note the format of the matrix that yields this obvious solution:

Whenever possible, aim for this format.

SOLVING SYSTEMS OF LINEAR EQUATIONS An equation is said to be linear if every variable has degree...

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