Jun 11
. SYSTEMS
1 OF LINEAR EQUATIONS
P
arameters
-
def A linear equation in variables is equation of the
formi EX : show that , for
any value of S and t that x E-s +
1 Xn = E + s + 2
Y
: n an ,
=
,
.
a ,
X, + aeXz +... + anXn =
b X3 = S , x = tpas
& infinitely
X, -
2xz + 3xz + Xy = 30 we us it for
many Sol
where X Xz , Xa are called "variables" and his a "constant" is a sol to the system 2x
,
-
xz +
3xz Xy -
= .
0
, ..., ,
"coefficients" (t - S 1) 2(t +
s + 2) + 35 t 3
verify
=
D
+ +
G1 Az An called Let's
-
and
-
are :
term , , ...,
t -
s+ 1 -
2t -
25 -
4 + 35 + t = -
3
3 3-
-2x 3x 4xz X4 1) = -
+
EX =
-
: + -
, =
X z 3 ② 2(t s +
1) (t + s 2) +
3s - t =
0
+
2y +
-
+ = -
·
· 2x +
3y =
43y = 33x + 443 2t -
25 + 2 - t -
S -
2 +
35 -
t =a
-
def : Given a linear equation a
,
x + auxz +... + anXn =
b
, 0 = c
S solutions if the : this is solution to the
the numbers .. Sz , .... Su are called a
system
equation is true when we substitute SiX ,,
Se = X2 , ...,
SniYn, ---
Equivalently if a . S ,
+
aSz +... + AnSa = b
Augmented matrix :
3x
+
2xz
-
Xz + xn = -
&
,
2Xi - x3 +
2xy = 0
5X4 We can write this
given 3x + x2 2x3 2
+ +
as
= :
b
XXH
*
X,
(o -20 (
I
I
-
3
EX : X +
y
+ z = 3
-"augmented" matrix
here is a solution :
x = 1 y = 1 , z =
,
-
1 + 1 + 1 =
3 How can we solve general system
:
here is another X = 3 , y = 0 ,
z = 0 The following elementary operations can be used :
1 .
interchange 2 equations
100
2x +
3y =
def A of equations collection ex [ = E x y 100
simply
:
system linear is + =
:
a
of linear equations .
2
Multiply an
equation by a nonzero number
ex :
G2 =
ex :
G . We
3 can add a multiple of 1
equation
to another
Def :
A solution to a
system of equations is a solution for ex E *
each of the equations at the same time
Ex Let's
E
:
operations to
Ex : x +
y
+ 3 = 3
>
- he wants us to verify this use elementry solve
S
2y 2
(d < / ;2) -
(a)
Sl
X + = -
+
2x 32 = 1 X -2 5 z =
y
+ +
y
=
here's a solution ,
=
,
2x +
y
=
7
umti
~
-centrati (x
+
2y =
2
observation :
A system of linear equations either has :
j -
-
1/3Rz
>
-
R2
. Unique solution
1 2.
infinitely-many 3 no.
solution
solution :
- - 16/3
↳ The goal X =
"consistant" "inconsistant" y =
-
1/3
-
100
x +
y =
Part 2 of class in
y X + 100
2
=
Y=
-
notebook
one intersection- uniquesal
-
infinately many no solution
. SYSTEMS
1 OF LINEAR EQUATIONS
P
arameters
-
def A linear equation in variables is equation of the
formi EX : show that , for
any value of S and t that x E-s +
1 Xn = E + s + 2
Y
: n an ,
=
,
.
a ,
X, + aeXz +... + anXn =
b X3 = S , x = tpas
& infinitely
X, -
2xz + 3xz + Xy = 30 we us it for
many Sol
where X Xz , Xa are called "variables" and his a "constant" is a sol to the system 2x
,
-
xz +
3xz Xy -
= .
0
, ..., ,
"coefficients" (t - S 1) 2(t +
s + 2) + 35 t 3
verify
=
D
+ +
G1 Az An called Let's
-
and
-
are :
term , , ...,
t -
s+ 1 -
2t -
25 -
4 + 35 + t = -
3
3 3-
-2x 3x 4xz X4 1) = -
+
EX =
-
: + -
, =
X z 3 ② 2(t s +
1) (t + s 2) +
3s - t =
0
+
2y +
-
+ = -
·
· 2x +
3y =
43y = 33x + 443 2t -
25 + 2 - t -
S -
2 +
35 -
t =a
-
def : Given a linear equation a
,
x + auxz +... + anXn =
b
, 0 = c
S solutions if the : this is solution to the
the numbers .. Sz , .... Su are called a
system
equation is true when we substitute SiX ,,
Se = X2 , ...,
SniYn, ---
Equivalently if a . S ,
+
aSz +... + AnSa = b
Augmented matrix :
3x
+
2xz
-
Xz + xn = -
&
,
2Xi - x3 +
2xy = 0
5X4 We can write this
given 3x + x2 2x3 2
+ +
as
= :
b
XXH
*
X,
(o -20 (
I
I
-
3
EX : X +
y
+ z = 3
-"augmented" matrix
here is a solution :
x = 1 y = 1 , z =
,
-
1 + 1 + 1 =
3 How can we solve general system
:
here is another X = 3 , y = 0 ,
z = 0 The following elementary operations can be used :
1 .
interchange 2 equations
100
2x +
3y =
def A of equations collection ex [ = E x y 100
simply
:
system linear is + =
:
a
of linear equations .
2
Multiply an
equation by a nonzero number
ex :
G2 =
ex :
G . We
3 can add a multiple of 1
equation
to another
Def :
A solution to a
system of equations is a solution for ex E *
each of the equations at the same time
Ex Let's
E
:
operations to
Ex : x +
y
+ 3 = 3
>
- he wants us to verify this use elementry solve
S
2y 2
(d < / ;2) -
(a)
Sl
X + = -
+
2x 32 = 1 X -2 5 z =
y
+ +
y
=
here's a solution ,
=
,
2x +
y
=
7
umti
~
-centrati (x
+
2y =
2
observation :
A system of linear equations either has :
j -
-
1/3Rz
>
-
R2
. Unique solution
1 2.
infinitely-many 3 no.
solution
solution :
- - 16/3
↳ The goal X =
"consistant" "inconsistant" y =
-
1/3
-
100
x +
y =
Part 2 of class in
y X + 100
2
=
Y=
-
notebook
one intersection- uniquesal
-
infinately many no solution
