MATH 110: Algebra Module 3 (Notes)

.1
3
Studying
of
Simultaneous equation in two variables, called
system equations.

Example (y4x + 2x = 3
-
:




-
-

yy
= = -

4

Ex Problem : The perimeter of a standard tennis court is 128 Ft
.
The width is 49ft less than the length. Find the
dimensions.
Let x width.
length y-
=
,




The perimeter is 128ft.
Y Y 128 x + + x +
y y
=

X
< 128 2x +
2y
=


the dimensions X,y The width is 49ftless than the
form the solution :
length: y =
X -
49


3
128
2x
2y
=
+

49
y X
= -




A solution of a equations in two variables
system of two is an
ordered pair
satisfying both equations simultaneously.

Example Problem : Consider the system
S2y y 3x-1 I
=



x+ =
7 #
a) is (5 7)
, a solution for the system ?
We have and =7
X 5
y .
=



Check for (1) 14= 14/
:
Zy = 3x 1 -




> (5 7)
207 3 5-1 = .

is a solution
,
14 15-1 =
of (E) .




Check for (11) X+y : = 7
5 +7 = 7 > ,7)
(5 is not a solution of (II) .




12 TX
Conclusion : (5 7) . is not a solution of the system .

, b Snow (3 4) , is a solution of the system.
X= 3, y 4=




Check for (1): Zy = 3 x-18 : 8/
2 4= 3 · .
3-1 ( (3,4) solves (1) .




8 = 9 1 -




Check for (*)
XYV
:


< (3 4) solves for
, (E) .




Conclusion (3 4) :
,
solves the system.


Graphing Method is NOT as reliable·


EX1
EY3x(#) (I)
:




↑




·...
(1) :
x int (1 0) :
,


y-int (0:
3) ,



(D) : -int (,
y-int: (0, -2)




In a
system
:
Sax +
by (1)
= r
(x +
dy S(π)
=



A solution has to be true for both equations.
When we graph the lines , we get :




1 The lines do not intersect (parallel).

Why? The system has no solutions. It is called inconsistent
.