Download Section 7-3

Section 7-3
Solving 3 x 3 systems of
equations
Solving 3 x 3 Systems




substitution (triangular form)
Gaussian elimination
using an augmented matrix (algebraic)
using an inverse matrix (calc.)
Substitution – Triangular Form

if you can simplify a system of
equations into the triangular form seen
below, you can solve the system by
substitution
x – 2y + z = 7
y – 2z = -7
z=3
Gaussian Elimination

this method is used to change a
3 equation-3 unknown system of
equations into triangular form
x – 2y + z = 7
3x – 5y + z = 14
2x – 2y – z = 3
x – 2y + z = 7

y – 2z = -7
z=3
Gaussian Elimination
The following operations can be used:
1. interchange any two equations
2. multiply (or divide) one of the equations
by a real number
3. add a multiple of one equation to any
other equation
Augmented Matrix
using an augmented matrix does the
same work as elimination without
having to re-write the equations and
variables over and over again
 the following is the augmented matrix
for the same system used earlier

1 2 1 7 
3 5 1 14 


 2 2 1 3 
Augmented Matrix
the goal is to use the same techniques
as for elimination to change the matrix
into “triangular form” so it can be
finished off with substitution
 “triangular form” for an augmented
matrix is called row echelon form

1 # # # 
0 1 # # 


0 0 1 # 
Row Echelon Form

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rows consisting of all 0’s are on the
bottom
the first non-zero entry of a row is 1
1’s are along the diagonal
the row echelon form of a system is not
unique
1 # # # 
0 1 # # 


0 0 1 # 
Row Operations
The row operations for a matrix are the
same as for Gaussian elimination
1. Interchange any two rows
2. Multiply all elements of a row by a
number
3. Add a multiple of one row to any
other row
Reduced Row Echelon Form
instead of stopping to use substitution
after you get to row echelon form, you
can keep going using the augmented
matrix to get the solution
 this new form is called reduced row
echelon form

1 0 0 a 
0 1 0 b 


0 0 1 c 
The solution of the
system is (a , b, c)
Order of Attack
17 06 04 a#
10 15 04 b#


10 02 13 c#
Special Cases


when using either technique and the
variables disappear from a row, it
leads to a special case:
if you are left with . . .
false statement = no solution
true statement = infinitely many
solutions