Intermediate Algebra - Reference Guides

434 views4 pages
4 Jun 2015
School
Department
Course
Professor
LINEAR EQUATIONS INEQUALITIES
l e a r n r e f e r e n c e r e v i e w
Intermediate Algebra
Intermediate Algebra
23
3211
324 1
323
3211
01110 32
223
07519
01110 32
23
0
32
21311
72
xyz
xyz
xyz
RR xyz
xyz
xy z
RR xyz
xyz
xy z
RR xyz
x
−+=
+−=
−+=
−+=
+−=
−+ =
−+=
+−=
−+ =
+
−+=
+77519
00
123
15
7
15
7
yz
xy z
zyx
−=
++ =
=− = =
34334
43 43
9510 8 0213 4
334
02135
0213 4
34
0213
31
21 32
xyz R R xyz
xy z xy z
xy z xy z
RR xyz
xy z
xy z
RR xyz
xy
+−= − +−=
++ = ++ =
++ = ++ =
+−=
++ =
++ =
+−=
++zz
xyz
=
++=
5
000 9
Zero term(s) on diagonal: No solution
Word Problem
• Two sums are to be invested in
parts yielding 4% and 5% return:
$40,000 to yield a total return of
$1800 and $12,000 to yield $200
• How should the investment be split?
ONE UNKNOWN
Word Problems
The sum of two numbers is 12 and one
number is three times the other
• What are the numbers?
x+ 3x= 12 4x= 12 x= 3
By how much should the radius of a circle
(r= lm) be increased to double the area?
Salt crystallizes from water when its
concentration reaches 50% How much
water should be evaporated from 40 kg of
a 20% solution to trigger crystallization?
0.5(40 – x) = 0.2 x 40 x= 24 kg
Two trains traveling at an average speed of
120 and 80 km/hr leave at a distance
300 km apart When will they meet?
120t+ 80t= 300 t= 1.5 hrs
Example
2(y + 3) = 5(y – 1) 7(y – 3) 4y = 10
2y + 6 = 5y – 5 7y + 21 y = 10/4 = 5/2
NUNKNOWNS
• Linear equations in n
unknowns can be solved
by Gaussian Elimination,
where the constant
coefficients of the
equations are arranged in
an array
• Various multiples of the
rows are added to or
subtracted from other
rows until zeros are
obtained above or below
the diagonal of the
coefficients
• The unknowns are then
calculated by successive
back-substitutions; if one
or more of the diagonal
elements is zero, then no
unique solution exists
• Linear equations in one unknown have the form ax + b = 0with the solution x = – b/a
TWO UNKNOWNS
• Two linear equations in two unknowns have the form: a1x+ b1y= c1a2x + b2y= c2
Solution Methods
By substitution: Solve for one unknown
(xor y) in either of the equations and
substitute in the other equation
24 24 1
23 348 2
xy y x x
xy x x y
−= =− =
+= =+ =
25 25 107
274 22574
xy y x
yx x x
−= =− −=
=+ − =+( ) no solution
xy=1
x+y=1
x
y
x+y=2
x
y
12
x+y=1
x+y=1
x
y
1
5x+5y=5
Consistent equations
have one solution:
y= 0, x= 1
Inconsistent equations
have no solution as
lines are parallel
Dependent equations
have an infinite
number of solutions
x + y = 40 000
0.05x + 0.04y = 1 800
x = 20 000
y = 20 000
x + y = 12 000
.05x + 0.4y = 200
x = –28 000
Therefore, no solution
Examples
By graphing: Plot
both equations
yielding two straight
lines • The point of
intersection represents
the solution If there
is no intersection,
then the solution does
not exist or there is
an infinite number of
solutions
ππ
() .121 2 1 0414
22
+= ⇒==xxm
DEFINITIONS
RULES OF OPERATION
x+y=1
xy=1 x+y=1
x+y=2 x+y=1
5x–5y=5
a > b ais greater than b
a < b ais smaller than b
a b ais greater than or
equal to b
a b ais less than or equal
to b
a > b, Inequalities having the
c > d same sense
b < a < c ais greater than b and
less than c
a <> b ais not equal to b
a > 0 ais a positive number
a < 0 ais a negative number
ıaı awithout sign =
absolute value of a
ıaı 0 ıaı is always greater
than or equal to zero
a > b, Inequalities having the
c < d opposite sense
b a c ais greater than or
equal to b and less
than or equal to c
Addition of the No
same constant change
on both sides
Subtraction of the No
same constant change
from both sides
Multiplication/ No
division of both change
sides by a positive
constant
Multiplication/ Inequality
division of both sign
sides by a negative reversed
constant
Raising each side No
(assumed positive) change
to a positive
number
Raising each side Inequality
(assumed positive) sign
to a negative reversed
number
Adding two No
inequalities of the change
same sense
Transfer of a term Term
to the other side changes
sign
Removal of –b <a<b
absolute sign
from |a| < b
TM
permacharts
© 1998-2012 Mindsource Technologies Inc.
INTERMEDIATE ALGEBRA • A-631-91www.permacharts.com
Unlock document

This preview shows page 1 of the document.
Unlock all 4 pages and 3 million more documents.

Already have an account? Log in

Get OneClass Grade+

Unlimited access to all notes and study guides.

Grade+All Inclusive
$10 USD/m
You will be charged $120 USD upfront and auto renewed at the end of each cycle. You may cancel anytime under Payment Settings. For more information, see our Terms and Privacy.
Payments are encrypted using 256-bit SSL. Powered by Stripe.