Warm-Up: Solve the system of
equations using Cramer’s Rule AND by
y
matrix inverses
Ex)
5x − 2 y = 5
x+ y =8
Thurs/fri 2/16 and 2/17
9 Warm up
9 Check HW,
HW stamp
9 Solving systems of equations using
Augmented Matrices (2.3/2.4)
9 Graphing calcs!
9 Start HW (2-4) Pg. 82
(#9 12 13 15) and
(#9,12,13,15)
d yellow
ll
graph
h
calc worksheet
Now let’s Solve the same system of
using Augmented Matrices
These are the “rules”:
#1) You can switch any rows
(it’s helpful to have a “1” in the
top left corner!)
#2) You can multiply any rows
by a nonzero #
0 x⎤
#3) You can add⎡1 rows
together
⎢
⎥
⎣0 1 y ⎦
#4) The final answer should be
the identity matrix with an
additional column ((the solution),
),
For ex:
5x − 2 y = 5
x+ y =8
You try…
Ex1)
4 x − 7 y = −2
x + 2y = 7
Let’s try a 3x3…
Ex 2)
x+ y+z =3
3 x − 2 y − z = −4
x + y − z = −1
Main Keys
we will be
using
Solve using Matrices and the TI
graphing calculator…
calculator
Ex)
5x − 2 y = 5
x+ y =8
Procedure
a) Write the (2x3, 3x4, 4x5,
etc) matrix on your paper.
b) Turn graphing calc ON
c) Select MATRIX (may be 2nd
& x-1)
d) Arrow to the right to select
EDIT. Hit enter.
Solve using Matrices and the TI
graphing calculator…
calculator
Ex)
5x − 2 y = 5
x+ y =8
Procedure
a) Write the (2x3, 3x4, 4x5,
etc) matrix on your paper.
b) Turn graphing calc ON
c) Select MATRIX (may be 2nd
& x-1)
d) Arrow to the right to select
EDIT. Hit enter.
e) Type matrix dimensions.
(2x3 for this example)
f) Fill in the elements…
Solve using Matrices and the TI
5 x − 2 y = 5 graphing calculator…
x+ y =8
Procedure
a) Write the (2x3, 3x4, 4x5,
etc) matrix on your paper.
b) Turn graphing calc ON
c) Select MATRIX (may be 2nd
& x-1)
d) Arrow to the right to select
EDIT. Hit enter.
e) Type matrix dimensions.
(2x3 for this example)
f) Fill in the elements…
Solve using Matrices and the TI
5 x − 2 y = 5 graphing calculator…
x+ y =8
Procedure
a) Write the (2x3, 3x4, 4x5,
etc) matrix on your paper.
b) Turn graphing calc ON
c) Select MATRIX (may be 2nd
& x-1)
d) Arrow to the right to select
EDIT. Hit enter.
e) Type matrix dimensions.
(2x3 for this example)
f) Fill in the elements…
g) Go to the home screen
Solve using Matrices and the TI
5 x − 2 y = 5 graphing calculator…
x+ y =8
Procedure
e) Type matrix dimensions.
(2x3 for this example)
f) Fill in the elements…
g) Go to the home screen
h) Select Matrix again. Go over
to Math.
Math
i) Go down to B: rref
j) Hit enter
Solve using Matrices and the TI
5 x − 2 y = 5 graphing calculator…
x+ y =8
Procedure
e) Type matrix dimensions.
(2x3 for this example)
f) Fill in the elements…
g) Go to the home screen
h) Select Matrix again. Go over
to Math.
Math
i) Go down to B: rref
j) Hit enter
k) Select Matrix again. Pick
your matrix (A) and hit
enter
Solve using Matrices and the TI
5 x − 2 y = 5 graphing calculator…
x+ y =8
Procedure
e) Type matrix dimensions.
(2x3 for this example)
f) Fill in the elements…
g) Go to the home screen
h) Select Matrix again. Go over
to Math.
Math
i) Go down to B: rref
j) Hit enter
k) Select Matrix again. Pick
your matrix (A) and hit
enter.
enter
l) Hit enter again.
Solve using Matrices and the TI
5 x − 2 y = 5 graphing calculator…
x+ y =8
I matrix
And
solutions!
Procedure
e) Type matrix dimensions.
(2x3 for this example)
f) Fill in the elements…
g) Go to the home screen
h) Select Matrix again. Go over
to Math.
Math
i) Go down to B: rref
j) Hit enter
k) Select Matrix again. Pick
your matrix (A) and hit
enter.
enter
l) Hit enter again.
Let’s try a 3x3…
Ex 2)
x+ y+z =3
3 x − 2 y − z = −4
x + y − z = −1
Procedure
e) Type matrix dimensions.
(3x4 for this example)
f) Fill in the elements…
g) Go to the home screen
h) Select Matrix again.
g
Go over
to Math.
a) Write the (2x3, 3x4, 4x5,
etc) matrix on your paper.
i) Go down to B: rref
b) Turn graphing calc ON
j) Hit
Hi enter
c) Select MATRIX (may be 2nd
& x-1)
k) Select Matrix again. Pick
your matrix (A)
y
( ) and hit
enter.
d) Arrow to the right to select
EDIT. Hit enter.
l) Hit enter again.
Wrapping it all up!
• Solving Systems of Equations
– Algebraically
g
y
• Substitution
• Elimination
• Or Graphing
– Using Matrices
• Inverses (X=A-1B)
• Cramer’s Rule (ratio of Determinants)
• Augmented Matrices (solution is last column in Imatrix)
• HWp.82 #9,12,13,15 use Augmented
Matrices now… Yellow Graph calc
worksheet.