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Partner Practice

Solve the linear system by graphing. Check your solution.

2x + y = 9 x - 3y = 6

2x + 3y = 15 2x - 3y = 3

-x + y = -2 2y + 4x = 12

2x – y = 6 2x – y = -10

 

Coll

 

 

 

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1

2

3

4

5

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2. a. W

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se

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5. Calculate

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osition0andonstofindwh

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xplainthem

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6  

7.2 Day 2 Solving Systems of Equations: Substitution

PartnerActivity!

1)SolvethesystemofEquationsbyGraphing:2)SolvethesystemofEquationsbyGraphing:

y x 7

x 4y 8

2x 3y 9

y x 7

ThinkandDiscuss:Didyouexperienceanyfrustrationswiththeproblemsabove?Describeinacouplesentencesbelow:

AlternativestoGraphing:SUBSTITUTIONMETHOD!

ThinkandDiscuss:WhatdoesthewordSUBSTITUTEmean?

SolvethefollowingusingtheSubstitutionMethod.

y x 7

x 4y 8

2x 3y 9

y x 7

 

Let’

Ex.1

Ex.3

Ex.6

St

1.

2.

3.

4.

5.

’sPractice

1

x 2y

x y 23

3

y 2x

y x

6Ginawents

coworkers.

boxesands

tepsforSUBS

.Decidewhich

._______________

._______________

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_______________

eSubstitut

73

x 19

7

shoppingfor

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spends$152

STITUTIONME

hvariableisea

______________fo

______________th

dvalueofone

_________bothv

tion

rholidaypre

ocolatecost$

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ETHOD!

asiestto______

orthatvariabl

heexpression

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valuesbyplug

sents.Sheb

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yboxesofcho

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________________

le.

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ndthevalueo

ggingthembac

Ex.2

Ex.

oughtboxes

boxesoforn

ocolatesand

hisappeartob

_________.

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oftheotherva

ckintotheori

2

7x 2y

4x y

4

y 2x

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ofchocolate

namentscost

howmanyb

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ariable.

iginalequatio

248

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?

8  

7.3 Solving Systems of Equations: Elimination

Example 1: The sum of two numbers is -5, and the difference of the two numbers is -17. What are the two numbers?

a) Set up an equation for the sum of two numbers.

b) Set up an equation for the difference of the two numbers.

c) Can you figure out how to solve to find the two numbers?

Example 2: Two small pitchers and one large pitcher can hold 8 cups of water. One large pitcher minus one small pitcher constitutes 2 cups of water. How many cups of water can each pitcher hold?

a) Set up an equation for the two small pitchers and one large pitcher.

b) Set up an equation for the one large pitcher and one small pitcher.

c) Use the two equations to find how many cups of water each pitcher can hold.

 

Examslide they

Examtakes

a

b

mple 3: Bill anand the giganhave the follo

a) Basedthe sy

b) How m

mple 4: On a ts longer than t

a) The flighthours longto Charlot

b) Let s be thwind. Uswind and

nd Steve decintic Ferris whowing tickets

d on what we ystem of linea

much does it c

typical day wthe return trip

t from Charlog. Find the avtte.

he speed (in me your answethe speed of t

ide to spend thheel. Their tic:

did for the laar equations b

ost to slide on

with light windp because the

otte to Phoenixverage speed (

miles per hourer to part (a) tothe wind.

he afternoon ckets are stam

ast two probleby elimination

n the Water S

ds, the 1800 mplane has to

x is 4 hours 3(in miles per

r) of the planeo write and so

at an amusemmped each tim

ems, set up twn.

Slide? How m

mile flight frofly into the w

30 minutes lonhour) of the a

e with no winolve a system

ment park enjome they slide o

wo equations (

much does it c

om Charlotte, wind. (Distan

ng, and the fliairplane on th

nd, and let w bm of equations

oying their faor ride. At th

(one for Bill a

ost on the fer

North Carolice = rate x tim

ight from Phohe way to Pho

be the speed (s to find the sp

avorite activithe end of the a

and one for S

rris wheel?

ina, to Phoenime)

oenix to Charoenix and on t

(in miles per peed of the pl

9

ies, the waterafternoon

teve) to solve

ix, Arizona,

rlotte is 4 the return trip

hour) of the lane with no

9 

r

e

p

10  

Example 4: Solve the linear system using elimination.

a. 145y4x

95yx

b. 262y3x

24y3x

You try!

c. 129y8x

123y8x

d. 86y7x

46y5x

Example 5: Solve the linear system using elimination (Arrange like terms)

a. 6x + 7y = 16 b. 4x – 5y = 5

y = 6x – 32 5y = x + 10

11  

7.4 Solving Systems of Equations: Elimination

Example 1: The play, Noises Off, costs $5 for students/seniors and $10 for adults. 500 tickets were sold for a total of $4,085. How many student/senior tickets were sold? How many adult tickets were sold?

Example 2: Jacob and Cody decide to go to Taco Bell for lunch. Jacob ordered 3 soft tacos and 3 burritos for $11.25. Cody ordered 4 soft tacos and 2 burritos for $10. How much does each soft taco cost? How much does each burrito cost?

Example 3: The Detroit Pistons are on a winning streak! Crazy, right? In one of their last games, they scored a total of 127 points in the game and 52 total baskets. If they made 11 free throws, how many 2 point and 3 point shots were made?

Example 4: Solve the linear system using elimination (multiply one equation, then add or subtract)

a. 3x – 3y = 21 b. 2x + y = -9 8x + 6y = -14 4x + 11y = 9

12  

Example 5: Solve the linear system using elimination (Multiply both equations, then add or subtract)

a. 7x + 2y = 26 b. 3y = -2x + 17

10x – 5y = -10 3x + 5y = 27

Partner Practice! Solve the linear system using elimination

1. 6x – 2y = 1 2. 2x + 5y = 3

-2x + 3y = -5 3x + 10y = -3

3. 3x – 7y = 5 4. 2x – 3y = 6

9y = 5x + 5 4y = -7x – 8

13  

7.5 Solving Special Types of Linear Systems Warm Up with your Partner: One admission to an ice skating rink costs x dollars and renting a pair of skates costs y dollars. A group pays $243 for admission for 36 people and 21 skate rentals. Another group pays $81 for admission for 12 people and 7 skate rentals. Determine the cost of admission and the cost of renting skates. (1-2) Warm Up: Graph the following systems of equations, then record your observations.

1. 2y x 4

32

y x 23

a. Where do your lines intersect?

b. How can you be sure?

c. What do you think the solution to the system is? Why?

2. 6x 8y 40

3x 4y 20

a. Where do your lines intersect?

b. How can you be sure?

d. What do you think the solution to the system is? Why?

14  

3. Discuss your findings above with your partner: then sketch a graph of a system with two lines that has:

a. one solution b. no solutions c. infinitely many solutions

Part I: a. Graph ANY two lines that are parallel. b. Write the equation of these two lines in slope intercept form.

c. Solve the system of equations using substitution. d. What do you discover? Explain why are you ending up with this solution.

Part II:

a. Solve the following system using elimination:

x + 3y = 15 -2x – 6y = -30 b. Based on the result above, what do you predict is the solution?

c. What would you predict this looks like graphically? d. Graph the system of equations.

x

yRecognizingSpecialCasesAlgebraically 

x

y

15  

Part III: Practice Solve the following systems algebraically and find the solution. Then state what the system looks like graphically.

a. 3x + 2y = 10 b. y = 7x + 4 -3x - 2y = - 2 -21x + 3y = 12

Part IV: Now What? Consider the system: 2

y x 43

3x y 5

a. Can you find the solution to the system by graphing? Why or Why not? b. Solve the system algebraically: c. Did you experience any frustrations solving algebraically? Could you have found that answer graphically?

Part V: Journal Questions: 1. Write a summary describing what you learned today about linear systems and their solutions (both graphically

and algebraically).

x

y

16  

Section 6.7: Graph Linear Inequalities in Two Variables

Linear inequality in two variables: replace the ‘=’ sign in a linear equation with <. , >, or .

Example 1: Tell whether the ordered pair is a solution of the inequality

a. 3x – 4y > 9 (2, 0)

b. 2x + 3y 14 (5, 2)

c. y 8 (-9, -7)

 

Step

Exam

a.

c.

ps for graph

Step 1: P

U

U

Step 2: D

S

S

mple 2: Gra

y < - 12

x

2y + 4x >

hing a linea

Put the ineq

se a ______

se a ______

Determine w

hade _____

hade _____

aph the ine

+ 4

> 8

r inequality

quality into

_______ line

_______ line

which side

________ the

________ the

quality

y in two vari

a nice grap

for < or >

for ≤ or ≥

of the line t

e line for < o

e line for > o

iables:

phing forma

to shade, a

or ≤

or ≥

b.

d.

ant, and th

and then sh

y 3x +

x + 4y <

hen graph t

hade that e

+ 1

- 8

he bounda

entire region

17

ary line.

n.

7 

 

e.

g.

 

y < 2

y > -2x +

+ 3

f.

h.

x 1

x 4

188 

 

Sect

Syste

____

Solu

____

Gra

____

____

Exam

y > -

y

tion 7.6: Sol

ems of linea

____________

tions of a sy

____________

ph of a syst

____________

____________

mple 1: Gra

-x – 2

3x + 6

lve Linear S

ar inequalit

___________

ystem of lin

___________

tem of linea

___________

___________

aph the syst

Systems of L

ies – ______

____________

near inequa

____________

ar inequaliti

____________

____________

tem of ineq

Linear Inequ

____________

___________

alities –_____

___________

ies –_______

___________

___________

qualities

ualities

___________

___________

___________

___________

___________

___________

___________

Exa

y < 3

y

___________

____________

____________

____________

____________

____________

____________

ample 2:

3x

-2x + 1

____________

___________

___________

___________

___________

___________

___________

___________

___________

___________

___________

___________

___________

___________

19

_____

_____

_____

_____

_____

_____

_____

9 

 

Exam

x + y

y < x

Try :

y < x

y

mple 3:

y 5

x + 3

Graph the

x – 4

-x + 3

system of i

nequalities

Exa

y >

x

3y <

x >

y

3x +

mple 4:

1

4

< 6x – 6

-2

4

+ 4y 24

200