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Algebra I Unit 2 Linear Relationships

Chapter 6 Systems of Linear Equations and Inequalities

Lesson 6-1 Graphing Systems of Equations

Objectives: I can determine the number of solutions a system of linear equations has, ifany.

I can solve systems of linear equations by graphing.

CCSS: A.CED.3, A.REI.6, MP.3, MP.8

Example 1: Number of Solutions

Use the graph to determine whether each system is consistent or inconsistent and if itis independent or dependent.

a. b.

o Consistent> inconsistent?

.

independent

Guided Practice 1: Number of Solutions

Use the graph to determine whether each system is consistent or inconsistent and if itis independent or dependent.

a.

b. Example 2: Solve by Graphing

Graph each system and determine the number of solutions that ithas. If it has one solution, name it.

a.

:Consistent lndepedent

Consistent independent

nxtb infinitely,

0--0,37 ply

••

-

••.

many

08#

- 4y= - 12

=4y= - 12 ( QD-4 =41/84

)

-43=1,2-4y= -20

( " 5)

-8×-44 =# =/ =/ 9=5

b. Guided Practice 2: Solve by Graphing

Graph each system and determine the number of solutions that ithas. If it has one solution, name it.

a.

b. Real-World Example 3: Write and Solve a System of Equations

Alex rode 20 miles last week and plans to ride 35 miles per week. David rode 50 miles last weekand plans to ride 25 miles per week. Predict the week in which Alex and David will have riddenthe same number of miles.

rgwioe'e¥⇒I¥i¥÷Iz.

.

::.

9-

••

_Zy=. x - z run

-

z =z=z y=k× -11

y=x -2 50101%5,1)•

.y=¥×t3 ••

•a

-2

:F #3 1

→6/-1

→

y= -2×-3

ory

pgP8try:FLo#3 ¥.

1 -5µ5i9#42 21-7

×

Guided Practice 3: Write and Solve a System of Equations

Joe and Josh want to buy a video game. Joe has $14 and saves $10 a week. Josh has $26 andsaves $7 a week. In how many weeks will they have the same amount?

Lesson 6-2 Substitution

Objectives: I can solve systems of equations by using substitution.I can solve real-world problems involving systems of equations by usingsubstitution.

CCSS: A.CED.3, A.REI.6, MP.2

Example 1: Solve a system by Substitution

Use substitution to solve the system of equations.

Guided Practice 1: Solve a system by Substitution

Use substitution to solve the system of equations.

€y=

-467*2-

,ZOTR

2×1-(-4×+12)=2 y=-

-

xp-2×+12=2- iz

-12T¥÷÷l¥11=31-37+10

y= -9+10-

n2×+5-(3×+10)=-1 |7×= - 51

@2×+15×+50=-1 F F

17×+50=-111€

-50 -50

Example 2: Solve and Then Substitute

Use substitution to solve the system of equations.

Guided Practice 2: Solve and Then Substitute

Use substitution to solve the system of equations.

Example 3: No Solution or Infinitely Many Solutions

Use substitution to solve the system of equations.

Guided Practice 3: No Solution or Infinitely Many Solutions

Use substitution to solve the system of equations.

a. b.

Oxus +29×=2y -3

w

#2C3) -3

m( zy - 3) + 5y=24 * 6-3

leg-9+51=24 x@Hy -9=24 -

+9 to

I

,y=3÷y@

=+3×+3 × 19×7855't'ts

- y=3X - 13 19×-76

4×+5-(3×-13)=11T9T9

4×+15×-165=1 ) @#y=3C4) -13

12 - 13

⇐

2×+29=8-

-42#2y=

8¥-4 Nosolution

a *Zx - (2×-3)=8 ( ) -2

[D 2¥+348 - > ply

No 6g - 8×= -2 manysolution

Real-World Example 4: Write and Solve a System of Equations

A nature center charges $35.25 for a yearly membership and $6.25 for a single admission. Lastweek it sold a combined total of 50 yearly memberships and single admissions for $660.50. Howmany memberships and how many single admissions were sold?

Guided Practice 4: Write and Solve a System of Equations

As of 2009, the New York Yankees and the Cincinnati Reds together had won a total of 32 WorldSeries. The Yankees had won 5.4 times as many as the Reds. How many World Series had eachteam won?

Lesson 6-3 Elimination Using Addition and Subtraction

Objectives: I can solve systems of equations by using elimination with addition.

I can solve systems of equations by using elimination with subtraction

CCSS: A.CED.2, A.REI.6, MP.7

- y- y

X= single admission Xty=50y= yearly membership 35.254+6.25×2660.50*55.050|zt× - 12

X= 50J -12*38

35.25g +6.25 ( 50 - g) 1660.50W

35.25g -1312.50-6.255-660.529g -1312.5=66 ;

2-95-348- Red 's worbdan.es Xty= 32 29 F

y=lZFiyanraiswosrerdie, 5¥40

Xt 5. 4×=3z* 5

6. 4×=3z4=27

Example 1: Elimination Using AdditionUse elimination to solve the system of equations.

Guided Practice 1: Elimination Using AdditionUse elimination to solve the system of equations.

Example 2: Write and Solve a System of EquationsFour times one number minus three times another number is 12. Two times the first numberadded to three times the second number is 6. Find the numbers.

Guided Practice 2: Write and Solve a System of EquationsThe sum of two numbers is –10. Negative three times the first number minus the second numberequals 2. Find the numbers.

Standardized Test Example 3:Use elimination to solve the system of equations.

\_ 3×-6453=18

÷y2=3% 3×+9900=1890

3×=- 72so%@

\ 4yt3( 6) = 22

6× = 36

6- 6- 4g + ( 8=22- 18

- 18@¥¥o

4X - 3y = 12 43 )

zxth.IE#oItt33EIotu

Xty=- to

→*¥÷E*y¥t5Ii¥o

A. B. C. D.

Guided Practice 3: Use elimination to solve the system of equations.

F. G. H. J.

Real-World Example 4: Write and Solve a System of Equations

A hardware store earned $956.50 from renting ladders and power tools last week. The storecharged customers for a total of 36 days for ladders and 85 days for power tools. This week thestore charged 36 days for ladders, 70 days for power tools, and earned $829. How much does thestore charge per day for ladders and for power tools?

Guided Practice 4: Write and Solve a System of Equations

Tamera and Addie are throwing a birthday party for their friend. Tamera invited 5 fewer friendsthan Addie. Together they invited 47 guests. How many guests did each girl invite?

4×+25-28( f- 1

-%xt3y=#§ 5y= 10

4×+24=248 y=2

4×±¥¥x¥z¥, @

8bt3c= 118bt3tI)=H

L ¥ &b-7c= c= . ,8bF3=¥3

- 4c=4 86=14A =4 js

@ b= 13g

Lptlapdrdfre 956,50=362+8513= powerpntoul @zq =362+703

956.5=36 Ltssp

- ¥- szq

= -36L - FOP 829 = 36L +7018€

=t5=¥gp- p¥8'50 829=361+59575

36 [ = 234

L #6.50

A=Addie's friends Att =47 z6T=5

T = Tamera 's friendsA - T

=5, zu

-26

- - T= -212A= 52¥262€

Lesson 6-4 Elimination Using Multiplication

Objectives: I can solve systems of equations by using elimination with multiplication.

I can solve real-world problems involving systems of equations.

CCSS: A.REI.5, A.REI.6, MP.1

Example 1: Multiply One Equation to Eliminate a Variable

Use elimination to solve the system of equations.

Guided Practice 1: Multiply One Equation to Eliminate a VariableUse elimination to solve the system of equations.

Example 2: Multiply Both Equations to Eliminate a VariableUse elimination to solve the system of equations.

Z (9) ty= 23

( *

yx×I¥IYa18+5-23-

-18-

×=- 9 II. y@

( # - |8r -2g - -26 z( 2)+29k¥3r -129=-4 6+zq

- heF÷n÷roizgI€

M 4fDt3y=8( )5 20×+15 ] . -40

( )3 ok - isy= -69 #+35¥,

W# 35-12

Z9×= . 29 5 5

-29 If@ y@

Guided Practice 2: Multiply Both Equations to Eliminate a VariableUse elimination to solve the system of equations.

Real-World Example 3: Solve a System of Equations

A fishing boat travels 10 miles down-stream in 30 minutes. The return trip takes the boat 40minutes. Find the rate in miles per hour of the boat in still water.

Guided Practice 3: Solve a System of Equations

A canoeist travels 4 miles downstream in 1 hour. The return trip takes the canoeist 1.5 hours. Findthe rate of the boat in still water.

Lesson 6-5 Applying Systems of Linear Equations

Objectives: I can determine the best method for solving systems of equations.

I can apply systems of equations.

CCSS: A.REI.6, MP.2, MP.4

# 53

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34×-0

Deft×=oy=X=

Boatspeed-dO-tzxtlzyjzycor_reytnsteed4O-5x-2suD3CzO-XtD24o-zxt2y30-2x-2y30poI2yxx-2d_X-l7.5m@DTj-BIaoIspeedC4-lxt.lD1.s

'

y= current speed

4=1.5×1.5y

6=1.5×+1 .5y

x=3' rsmeh 4,5155×151

Example 1: Choose the Best Method

Determine the best method to solve the system of equations. Then solve the system.

Guided Practice 1: Choose the Best Method

Determine the best method to solve the system of equations. Then solve the system.

a.

b.

(

#z-4¥-6*-46-

\4×+4=34

- 2×+313=23

-H2y=- 12 2×+9=23

Ty If - q - q@ ¥¥#

FA IE¥¥k in

.is#y=9-7x=7=7=72t7y=9@7yj@ijIni5x' '

→¥⇒

→→y=5x#T - y=5( 3) - 17

as' 3×+2 (5×-17)=5 y=l5 - 17

3×+10×-34=5yd

l3×=3913×-34=5 T3 I

+34 +34 ×#

Real-World Example 2: Apply Systems of Linear Equations

Ace Car Rental rents a car for $45 and $0.25 per mile. Star Car Rental rents a car for $35 and$0.30 per mile. How many miles would a driver need to drive before the cost of renting a car atAce Car Rental and renting a car at Star Car Rental were the same?

Guided Practice 2: Apply Systems of Linear Equations

Jared has volunteered 50 hours and plans to volunteer 3 hours in each coming week. Maddie is anew volunteer who plans to volunteer 5 hours each week. Write and solve a system of equationsto find how long it will be before they will have volunteered the same number of hours.

Lesson 6-6 Systems of Inequalities

Objectives: I can solve systems of linear inequalities by graphing.

I can apply systems of linear inequalities.

CCSS: A.REI.12, MP.1, MP.6

Example 1: Solve by graphingSolve the system of inequalities by graphing.

O -O

⇒hfiogkf y= .25×+45

Y= .30×+35

.25×+45

= .30×t}s5-

. 25 ×- 35 -

.28x

t.IE?fsxX=2oomi@T=tofalhwrsT=3Wt50

We WeeksT= 5W 2w=50

5w=3w+5ow=2Ee⇐-3W

- 3W

Th.

a

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test# ( QD

0<20 ) +2

B, 0<2 True !

.

02 - 0 -3

02-3 True !

Guided Practice 1: Solve by graphingSolve the system of inequalities by graphing.

a.

b.

Example 2: No Solution

Solve the system of inequalities by graphing.

Guided Practice 2: No Solution

Solve the system of inequalities by graphing.

B17

•-→A lotintradayA Test 1710,0 ) 9

D

He0=3 True

1 ° 0+0>-1 False

Test Pt L 010 ) a

uDto0EIme '. •

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.

•

.

y< -2×+4 a..

-

2× - 2 : B

←→

13 Test Pt ( 010 ) →

. •

02-30+1 ← . .→

NO0>-1 False ! . • gowtimd. .

→

01 - 3( 0 ) -2

02=2 False !,

Real-World Example 3: Whole-Number Solutions

A college service organization requires that its members maintain at least a 3.0 grade pointaverage and volunteer at least 10 hours a week.

a. Define the variables and write a system ofinequalities to represent this situation. Then graphthe system.

b. Name one possible solution.

Guided Practice 3: Whole-Number Solutions

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The Theater Club is selling shirts. They have only enough supplies to print 120 shirts. They willsell sweatshirts for $22 and T-shirts for $15, with a goal of at least $2000 in sales.

a. Define the variables, and write a system of inequalities to represent this situation. Thengraph the system.

b. Name one possible solution.

c. Is (45,30) a solution? Explain.

1

-