Systems of Linear Equations Section 8.1

Graph and find the solutions to the systems of equations:

1.¯®­

�

3932

xyx

2. ¯®­

� �

2313

xyxy

3. ¯®­

� �

4242

xyxy

In exercises 1 – 4, solve the system of equations by GRAPHING.

yxyx

2423

-==+

2131274� ��

�yx

yx

52234

�� �

yxyx

SUBSTITUTION METHOD

X Y Substitute the X yx 2-= 423 =+ yx ( ) 423 =+ y � �,

� �,

52234

�� �

yxyx

21912534� ��

�yxyx

ELIMINATION/ADDITION METHOD 242723

� �

yxyx

Substitute: Multiply: 242 � yx

� �� �242

723 � �

yxyx

� �, Substitute: Multiply: � �,

In exercises 1 – 8, solve the system of equations using the elimination method.

1) 4 7 2) 2 5 2 3) 2 5 4) 3 7 2 2 3 3 8 2 5 4

x y x y x y x yx y x y x y� � � � �� � � 2 3 14

5) 3 4 6) 4 3 7) 3 2 8) 5 2 2 7 8

x y

x y x y x y x yx y x y

�

� � � �� � 6 5 7 y x y x � �

Now try: xyx

yyx� �� �

93133

D = R * T

D = R * T

Applications (just set up the equations for the following)

Mixture Problems How many liters of a 14% alcohol solution must be mixed with 20L of a 50% solution to get a 30% solution?

How much pure dye must be added to 4 gal of 25% dye solution to increase the solution to 40%?

Uniform Motion (d=rt) A crew of eight can row 20 kilometers per hour in still water. The crew rows upstream and then returns to its starting point in 15 minutes. If the river is flowing at 2 km/h, how far upstream did the crew row? upstream downstream

�

�

14% 504C 30%

× 20 yEe : × + 20 = y[ ; 14×+206 ) =

3oY 14×+1000=30 y

100%25%4040× 4 Y

E,

:X + 4 = yEz : loox +

loot0

2 D 1 8 15 - t

20

2 D 22t.pe#D= iras . t )

D= 270 - 18 t

D

÷22T

� �

� �

D = R * T

D = R * T

A plane carries enough fuel for 20 hours of flight at an airspeed of 150 mph. How far can it fly into a 30 mph headwind and still have enough fuel to return to its starting point? (This distance is called the point of no return) against wind with wind

Investment Melissa Wright won $60,000 on a slot machine in Las Vegas. She invested part at 2% simple interest and the rest at 3%. In one year she earned a total of $1,600 in interest. How much was invested? Invested Interest Blake has a total of $4000 to invest in two accounts. One account earns 2% simple interest, and the other earns 5% simple interest. How much should be invested in both accounts to earn exactly $155 at the end of 1 year? Invested Interest

% 30 D 120 t

30 D 180 20 - t

D= hot 120T = 3600 - ( settire tint

D= 3600 - 180T zcot =36WD= 1204427440€ t= -12

0<02

0.03×+11=60,

.coX Y 1600

0.02×+0.04=16002×+34=169000

0,02 0.05

× Y 4000 X Y 155

0.02×+0.054=1552×+54=15500

At $1.40 per bushel, the daily supply for soybeans is 1,075 bushels and the daily demand is 580 bushels. When the price falls to $1.20 per bushel, the daily supply decreases to 575 bushels and the daily demand increases to 980 bushels. Assume that the supply and demand equations are linear. A) use the supply data to form a supply equation. Hint:( Price , supply) ( , ) ( , ) B) Use the demand data to form a demand equation. Hint:( Price , demand) ( , ) ( , ) c) Find the equilibrium price and the quantity. A company produces Italian sausages and bratwursts at plants in Green Bay and Sheboygan. The hourly production rates at each plant are given in the table. How many hours should each plant be operated to exactly fill and order for 62,250 Italian sausages and 76,500 bratwursts?

Plant Italian sausage Bratwurst Green Bay 800 800 Sheboygan 500 1000

p s

m= 1075 - s >s(1.4/1075) ( 1,20,s>s )

¥2 =5o÷=2soo

Ytmxtb y=zsoo× - zqzs. m= 2500

515£,}sgh÷ytbS=zsoop-zq.(1.4/5844.2,ssv)

M=580 - 980 M= - zooo

¥2 -5400.07=-2000 D= . zooopt3380Y=mxtb580=-2404,4 ) tb

3380= b

2500 p - 2425 = - zoooptezzgo•$29,80044500ps = 5805

p = $1.29 D= - 20009.293+3280=800

:÷ y

62,250 76,500

800×+5004=62250

.

800×+1004=56,sco

x y z

Systems of three equations Section 8.2

542924

1352

� �� ��

� ��

zyxzyxzyx

Multiply:

542924

1352

� �� ��

� ��

zyxzyxzyx

924

1352 ��

� ��zyxzyx

542

924� ��

��zyxzyx

Back Substitute:

.

*'s'zI¥i ¥eF*÷¥syt

zx - syt3z- - 1 Y= 2- 2( ) -¥-8yt4z-=

13×+72=-19

x , ¥YzYIEE÷+22¥z 6ytZz=14

× - ZY - 4Z⇒ 614+22=14X - 2(z ) -44=-5 12+22=14

X - 4 - 4=-5 ZZ = z

X - 8=-5 Z =/×=z

(331 )

In exercises 9 – 24, solve the system of equations by any method.

9) 4 10) 2 3 5 1 11) 6 12) 5 4 6 2 2 5 4 2 4 3 4 2 1

x y z x y z x y z x y zx y z x y z x y z� � � � � � � � �� � � � � � 4 5 2

4 5 1 3 2 4 6 2 5 3 3 2 3 2

13) 3 5 2 8 14) 3 2 5 15) 3 4 13

x y zx y z x y z x y z x y z

x y z x y z x y z

� � � � � � � � � �

� � � � � � 16) 4 3 12 3 2 5 4 3 4 4 2 1 3 2 4 10 3 2 11 2 3 5

x zx y z x y z x y z x yx y z x y z x y

� �� � � � � � � � � �� � � � � � 2 7 4 7

17) 2 3 2 18) 2 4 6 19) 3 4 13 20) 2 2 3 2 6 2 13 6

z y z

x y z x y z x y z x y zx y z x y z x

�

� � � � � � � � � � �� � � � 2 12 2 5 6

3 3 10 2 2 6 10 4 3 7 3 2 7

1 2 3 1 1 121) 9 22) 4

y z x y zx y z x y z x y z x z

x y z x y z

� � � � �� � � � � � � � �

� � � � 1 1 3 1 2 1 23) 6 24) 1

2 1 2 5 1 1 2 2 5 5 4 1 8 0 1 1

3 1 4 3

a b c a b c

x y z x y z a b c a b c

x y z

� � � �

� � � � � � � � � �

� � 3 4 2 5 2 4 3 5 2 1 4 0x y z a b c a b c� � � � � � �

:a y=tbz=t

1×+2 ytts5×+4 ytz = -1

3 X tSy+2z=o

Infinitely many solutions or no solution?

𝟐𝒙 − 𝟑𝒚 = 𝟒𝟒𝒙 − 𝟓𝒛 = 𝟏𝟎

−𝟏𝟐𝒚 + 𝟏𝟎𝒛 = −𝟒

Special, but same principles: 1141523

�� ��

zyxzyx

solve in terms of z � �, , z

Try: 𝑥 − 𝑦 + 𝑧 = 1

2𝑥 + 𝑦 + 𝑧 = 67𝑥 − 𝑦 + 5𝑧 = 15

--

lzytlozt -4

' *:*:* '¥gIaF÷;E÷÷÷p¥¥¥I#\ 2×-4=4-4×+64=-8 -34=-2×+46*-2 ¥55

y=±3× -

4¥Sz=,o.s±÷¥¥E× Y

Ix-

2y+z=i€€,Z)f¥pts. 3(×t4y- 2- =/ , )

*ii¥'±¥÷f¥x¥E¥÷¥o- Ky .

-- tz - ( g ¥z[g,# 54 Fy

7×= - z +41Y=

}ztI×=

- fz+4f

Application Nina's nut shop produces a deluxe blend of almonds, cashews, and pecans. How many pounds of each nut should be used to produce 940 pounds of the deluxe blend at a cost of $4,300 if the blend is 50% cashews? Almonds cost $5 per lb. Cashews cost $4 per lb. Pecans cost $5.40 per lb. Try: Mega toys produces toys from three shops. Shop one requires $25 per 100 toys for employee wages, $20 per 100 toys for electrical costs, and $50 per 100 toys for materials. Shop two requires $35 per 100 toys for employee wages, $10 per 100 toys for electrical costs, and $40 per 100 toys for materials. Shop three requires $10 per 100 toys for employee wages, $20 per 100 toys for electrical costs, and $100 per 100 toys for materials. The total costs for employee wages, electrical costs, and materials in order are $16,500, $9000 , $32,000. How many toys are made at each shop?

cost #of1bs Total Cost

Almonds 5 X SXcashews 4 Y 4yPecans 5.4 Z 5.4 z

Total 9404300

Xtytz = 940

5×+44+5,42=430y=So%( 940)

y=47÷ E

Shopl s4cp2 stop ]' Total

bases 25 35 10 16500

costs 20 10 zo soco

matey SO 40 100 32,

.co

25×+35×+182=16,50020×+10×+20 Z = socu

50×+40×+1002--32,000X=?o0 y=}0o z=1Oo

200.00 39000 ( qoco

SYSTEMS OF LINEAR INEQUALITIES

Section 8.3 1) Graph the solution region. 2) Is the solution region bounded or unbounded. 3) Find the coordinates of the corner points.

𝟐𝒙 + 𝒚 ≤ 𝟖𝒙 + 𝟑𝒚 ≥ 𝟏𝟐

𝒙 ≥ 𝟎𝒚 ≥ 𝟎

312 t�� yandxy'' ¥5"3÷Yxt '

=

1¥)

.

.•g:*;•|B&•(

1,3 )

corner points' t Co ,HCq4)E¥

1) Graph the solution region. 2) Is the solution region bounded or unbounded. 3) Find the coordinates of the corner points.

𝒙 + 𝟑𝒚 ≤ 𝟏𝟖−𝒙 + 𝒚 ≥ 𝟐

𝒙 + 𝟑𝒚 ≤ 𝟏𝟖−𝒙 + 𝒚 ≥ 𝟐

𝒙 ≥ 𝟎𝒚 ≥ 𝟎

𝒙 + 𝟑𝒚 ≤ 𝟏𝟖

−𝒙 + 𝒚 ≥ 𝟐𝒙 ≥ 𝟎𝒚 ≥ 𝟓

3¥t¥t¥Y E -

tsxtS )• ( 0,6 )

or ( o, 2)

oi3¥)

. ( 0,6)

B or