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 ﬁ4x= 12 ﬁx= 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 ﬁx= 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 ﬁ t= 1.5 hrs

Example

2(y + 3) = 5(y – 1) – 7(y – 3) ﬁ4y = 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

x–y=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

ﬁx = 20 000

y = 20 000

x + y = 12 000

.05x + 0.4y = 200

ﬁx = –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

x–y=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

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