Name:_______________________________________________Date:_________________Per:________+

Unit%6:%%System%of%Equations%Notes%

Day%1+–+Solving%Linear%Systems%by%Graphing%

System%of%Linear%Equations%(Linear%System)%-++++ + + + +

+ + + + + + + + + + + + + +

Solution%of%a%System+-++ + + + + + + + + + +

Because+the+solution+of+a+linear'system+satisfies+each+equation+in+the+system,+the+solution+must+lie+on'the'graph'of'both'equations.+++

Use+the+graph+to+find+the+solution+of+the+linear+system.+

! = 2! − 4++ + ! = −! + 2+Where+do+the+lines+intersect?+

+

It+is+possible+to+have+3+different+types+of+solutions+when+solving+linear+systems.++

When+two+lines+intersect+at+_______________________+then+there+is+__________________+solution.++

When+two+lines+____________+intersect+they+have+__________+solution.+These+lines+are+_______________+

which+means+they+have+the+____________+slope.++

When+two+lines+intersect+at+___________________+then+there+is+____________________+solutions.+These+

lines+are+__________________+which+means+they+have+the+same+__________+and+______-intercept.+

Number%of%solutions%of%a%Linear%System+

%%%%%%%%%%%% % %

+++ +++++++++Solution+ +++++++++ + +Solution+ + ++++ ++++++++Solutions+

Identifying%the%Number%of%Solutions+

If+each+variable+equals+a+number,+there+is++ + + + .+

If+a+number+equals+itself,+there+are+'' ' ' ' .+

If+a+number+equals+a+different+number,+there+is+' ' ' ' .+

Example:+

y = 12x +1

y = 3x − 2+ + + +

y = 2x − 52y = 4x −10

+ + +y = 4x −1y = 4x + 4

+

%

Solving%a%Linear%System%using%Graph%and%Check+

One+method+of+solving+a+_______________________________+is+to+graph+the+equations+carefully+on+the+____________+coordinate+grid+and+find+their+point+of+intersection.++This+point+is+the+____________________+of+the+system.+

Step%1:%%++ + +each+equation+in++ + + + + + +form.+

Step%2:%%++ + +both+equations+on+the+same++ + + + + .

Step%3:%%+ + + +the+coordinates+of+the++ + + + + .+

Step%4:%%++ + +the+coordinates+algebraically.++(Plug+the+point+into+each+

+ +++ + + +of+the+system.)+

Find%the%solution%to%the%system%of%equations%

a)++y+=+-3x++10+ + + + + + + b)++y+=+x+++2+

+++++y+=+x+–+2+ + + + + + + + +++++y+=+(1/2)x+

+

+

+

+

+

+

+

Solution:+_________________+ + + + Solution:+_______________+

Chapter 5 8 Glencoe Algebra 1

Skills PracticeGraphing Systems of Equations

NAME ______________________________________________ DATE______________ PERIOD _____

5-1Copyright ©

Glencoe/M

cGraw-Hill, a division of The M

cGraw-Hill Com

panies, Inc.

Use the graph at the right to determine whethereach system has no solution, one solution, or infinitely many solutions.

1. y ! x " 1 2. x " y ! "4y ! "x # 1 y ! x # 4

3. y ! x # 4 4. y ! 2x " 32x " 2y ! 2 2x " 2y ! 2

Graph each system of equations. Then determine whether the system has nosolution, one solution, or infinitely many solutions. If the system has one solution,name it.

5. 2x " y ! 1 6. x ! 1 7. 3x # y ! "3y ! "3 2x # y ! 4 3x # y ! 3

8. y ! x # 2 9. x # 3y ! "3 10. y " x ! "1x " y ! "2 x " 3y ! "3 x # y ! 3

11. x " y ! 3 12. x # 2y ! 4 13. y ! 2x # 3x " 2y ! 3 y ! " x # 2 3y ! 6x " 6

x

y

Ox

y

O

x

y

O

1$2

x

y

Ox

y

Ox

y

O

x

y

Ox

y

Ox

y

O

x

y

Oy ! "x # 1

x " y ! "4 2x " 2y ! 2

y ! 2x " 3

y ! x " 1

y ! x # 4

Day$2$&$Solving$Linear$Systems$by$Substitution$

!

Last!class,!we!learned!that!on!a!graph,!the!solution!of!a!system&of&equations!is!where!the!_______________________________.!

Another!way!to!solve!a!system!of!equations!is!to!use!substitution.!

Substitution!means!to!_____________!one!value!with!an!______________________!value.!

Ex.!1!Substituting!a!value!

1.!x!=!3,!find!3x!+!7! ! 2.!x!=!!!;!2x!+!8! ! ! 3.!x!=!y,!!solve:!3y!+!2x!=!7!

!

!

!

A!simple!system!is!one!where!one!_____________________!is!already!solved!for.!All!we!have!to!do!

is!____________________________!the!value!into!the!unsolved!equation!and!solve!for!the!

________________!variable.!We!can!write!the!solution!to!our!system!as!an!_________________________!

of!(x,y).!

Ex.!2!Solving!a!simple!system!with!substitution!

1. x + 3y = 12x = 2

!! ! 2.!4x + 5y = 31y = 3

! ! 3.!3x − 7y = 212y = 6

!

!

!

Many!times!we!are!not!given!only!a!number!to!substitute!but!rather!an!______________.!The!

important!thing!to!remember!is!if!you!_________________!with!a!coefficient!we!must!always!

_______________!the!coefficient.!!

Ex.!3!Substitute!the!given!information.!

1.!x + 3y = 15x = 2y

! ! ! 2.!x + 3y = 18x = 2 + y

! ! 3.!2x + 3y = 12x = y +1

! 4.!x − 3y = 11y = x +1

!

!

!

After!we!have!replaced!and!_________________!our!system!we!need!to!____________!for!the!

variable!in!the!substitution!equation,!then!________________!our!solution!in!the!second!

equation!to!_____________!for!the!remaining!variable.!

Ex.!4!Solve!the!system.!

1.!x + 3y = 15x = 2y

! ! ! 2.!x + 3y = 18x = 2 + y

! ! 3.!2x + 3y = 12x = y +1

! 4.!x − 3y = 11y = x +1

!

!

!

!

More!complex!systems!require!us!to!_________________!for!one!of!the!variables!before!

substitution.!It!is!important!to!know!which!____________________!it!will!be!easier!to!solve!for.!

The!key!is!we!____________________!solve!for!the!variable!with!a!________________________!of!1.!

Ex.!5!Decide!which!equation!to!use!and!which!variable!to!solve!for.!

1. x + 2y = 11x − y = 2

!! ! 2.!−x + y = 23x − 2y = 3

!! ! 3.!x − y = 4x + y = 6

!

!

&

&

!Solve!the!above!for!your!chosen!variable:!

1.! ! ! ! 2.! ! ! ! 3.!

!

!

!

Finish!solving!the!system:!

!

!

!

Don’t!forget!!It!is!possible!for!a!system!to!have!_____________,!_______________!or!

________________________!solutions.!If!each!variable!equals!a!number,!there!is!! ! !

! .!If!a!number!equals!itself,!there!are!& & & & & .!If!a!number!

equals!a!different!number,!there!is!&& & & .!

Ex.!6!Solve!the!following!system!if!the!system!does!not!have!exactly!one!solution!state!

whether!it!is!no&solutions!or!infinitely&many&solutions.&

a) y!=!2x!+1!! ! ! ! ! ! b)!W4x!+!y!=!11!3x!+y!=!W9! ! ! ! ! ! !!!!!!y!–!4x!=!W13!

!

!

!

!

c)!!3x!–!y!=!15!! ! ! ! d)!2x!+!y!=!4!

!!!!!3x!=!15!+!y!! ! ! ! !!!!Wx!+!y!=!1!

!

!

!

!

Application:$$A!store!sold!a!total!of!125!car!stereo!systems!and!speakers!in!one!week.!!The!stereo!systems!sold!for!$104.95,!and!the!speakers!sold!for!$18.95.!!The!sales!from!these!two!items!totaled!$6926.75.!!How!many!of!each!item!were!sold?!

Define!the!variables.!

!

Write!equations!to!represent!total!items!sold!and!total!sales.!

!

!

Use!substitution!to!determine!how!many!speakers!and!car!stereos!were!sold.&

!

Name: ___________________________ Date: ___________ Period: ________

6 – 2 IN-Class Solving Systems by Substitution Solve the following systems.

1. 3y = 12 + xx = 3

2. 3x + 2y = 21y = 3

3. x + 2y = 8y = 5

4. x + 2y = 82x = 8

Solve the following systems:

5. y = 4x3x − y = 1

6. x = 2yy = x − 2

Decide which equation and which variable would be easiest to solve for:

7. x − 2y = −5x + 2y = −1

8. x + 2y = 03x + 4y = 4

Solve the following systems, if there is not one exact solution state whether it is no solutions or infinitely many solutions.

9. x = 2y − 3x = 2y + 4

10. c − 4d = 12c − 8d = 2

Day$3$Elimination)by)Addition)and)Subtraction)

To$Eliminate$means$to$get$___________$of$one$of$the$variables.$

When$we$_________$a$system$of$equations$by$__________________$we$are$eliminating$________$of$

the$variables$so$we$can$solve$for$the$other$one.$$

Remember$that$the$_______________$to$a$system$of$equations$is$the$ordered$pair,$or$_____$and$

______$value,$that$is$a$solution$to$both$equations.$$$

When$we$use$elimination$we$first$need$to$make$sure$our$variables$are$lined$up$so$that$one$

of$the$variables$can$be$_________________.$

Ex.$1$ReDarrange$the$variables$so$that$one$can$be$eliminated$

1.$5b + 2c = 108c − 5b = 20

$ $ $ 2.$5t + 2r = 103t + 2r − 8 = 0

$ $ $ 3.$3x + y = 7y − 5x = 13

$

$

$

$

After$all$our$variables$are$arranged$correctly$we$need$to$decide$if$we$want$to$_______$or$

______________$our$equations$to$________________$one$of$the$variables.$

Ex.$2$Decide$if$we$are$going$to$add/subtract$and$what$variable$will$be$eliminated$

1.$5b + 2c = 108c − 5b = 20

$ $ $ 2.$5t + 2r = 103t + 2r − 8 = 0

$ $ $ 3.$3x + y = 7y − 5x = 13

$

$

$

$

When$we$are$adding/subtracting$our$equations$we$need$to$make$sure$we$add/subtract$

______________$variables.$Once$you$add/subtract$one$of$your$variables$should$be$eliminated$

(__________).$We$then$________________$for$the$_________________________variable$and$then$

substitute$our$solved$value$into$one$of$the$__________________$equations$to$find$the$other$

variable.$Then$write$your$solution$as$an$____________________$pair.$$

Ex.$3$Solve$the$following$systems$

1.$5b + 2c = 108c − 5b = 20

$ $ $ 2.$5t + 2r = 103t + 2r − 8 = 0

$ $ $ 3.$3x + y = 7y − 5x = 13

$

)

)

)

)

)

REMEMBER:)

Step)1:)____________$the$equations$so$_________________$are$lined$up.$$$

Step)2:)_________$or$______________$the$equations$to$________________$one$of$the$$$$$$$$________________.))Use$______________$if$the$coefficients$have$different$signs$and$_________________if$the$coefficients$have$the$same$sign.))$

Step)3:)____________$the$equation$for$the$remaining$variable.$$$

Step)4:))) ) ) )the$value$obtained$in$Step$3$into$either$one$of$the$original$equations$and$solve$for$the$other$$ $ $ .)

Step)5:))$ $ $the$solution$as$an$_________________________________.$$$

Don’t$forget!$It$is$possible$for$a$system$to$have$_____________,$_______________$or$

________________________$solutions.$If$each$variable$equals$a$number,$there$is$$ $ $

$ .$If$a$number$equals$itself,$there$are$! ! ! ! ! .$If$a$number$

equals$a$different$number,$there$is$!! ! ! .$

Ex.$4$Solve$the$following$systems$

1.$3b + 2c = 10−3b + 3c = 5

$ $ $ $ $ 2.$5t + 2r = 102r + 5t = 0

$$ $

$ $

$

$

$

3.$5x + y = 3y − 2x = 18

$ $ $ $ $ $ 4.$3b + c = 18−c − 3b = −18

$

$

$

$

$

$

$

Example)5:)Negative$three$times$one$number$plus$5$times$another$number$is$D11.$$Three$times$the$first$number$plus$seven$times$the$second$number$is$D1.$$Find$the$numbers.$$$

)

)

)

)

)

Example)6:))The$sum$of$two$numbers$is$D10.$$Negative$3$times$the$first$number$minus$the$second$number$equals$2.$$Find$the$numbers.$$$

$

$

$

$

)

Example)7:)$Flying$to$Africa$with$a$tailwind$a$plan$averaged$271$m/h.$On$the$return$trip$the$plan$only$averaged$203$m/h$while$flying$black$into$the$same$wind.$Find$the$speed$of$the$wind$and$the$speed$of$the$plan$in$still$air.$$$$

6"3$Elimination$In$Class!! ! Name:_________________________________!! ! ! ! ! ! Date:___________________!Class:________!!Decide!if!you!would!use!addition/subtraction!to!solve!the!following!systems!and!what!variable!would!be!eliminated:!

1.!!!!!2t + 5r = 69r + 2t = 22

! ! ! ! ! 2.!!!!−4x + 3y = −34x − 5y = 5

!

!!!Solve!the!system!by!elimination!!

3.!!!!!2t + 5r = 69r + 2t = 22

! ! ! 4.!!!!−4x + 3y = −34x − 5y = 5

! ! 5.!!!!4y + 3x = 223x − 4y = 14

! !

!!!!!!!

6.!!!!!8b + 3c = 118b + 7c = 7

! ! !7.!!!!!

5m − p = 75m − p = 11 !! ! 8.!!

8x + 5y = 4−5y − 8x = −4

!

!!!!!!!!!9.!!The!sum!of!two!numbers!is!24.!!Five!times!the!first!number!minus!the!second!number!is!12.!!What!are!the!two!numbers?!!!!!!!10.!!Flying!to!Neverland!with!a!tailwind!a!plan!averaged!285!m/h.!On!the!return!trip!the!plan!only!averaged!219!m/h!while!flying!black!into!the!same!wind.!Find!the!speed!of!the!wind!and!the!speed!of!the!plan!in!still!air.!!!!

Day$4$Elimination$Using$Multiplication$$Sometimes(our(____________________(are(not(_______________(and(we(have(to(_______________((

to(create(opposites(before(we(can(use(___________________.((((

(

To(create(opposites(we(need(to(________________(one(of(our(equations(by(a(____________(

to(make(one(of(our(___________________(have(opposite(coefficients.(

When(we(look(at(our(system(we(need(to(decide(for(which(variable(will(it(be(easier(to(

create(_________________(coefficients.(To(decide(which(equation(to(multiply(first(

compare(coefficients(of(_________(variables(and(decide(which(___________________(you(

can(multiply(to(be(the(____________(and(then(what(__________(to(use(to(make(it(the(

__________________.((

Ex.(1(Decide(which(equation(to(multiply,(and(what(to(multiply(it(by.((

1.2x + 3y = 6x + 2y = 5

((( ( ( 2.(4s − t = 95s + 2t = 8

((( ( 3.(6x − 4y = −85s + 2t = 8 $ $

$

$ $

After(we(have(figured(out(what(to(multiply(one(of(our(equations(by,(we(______________(

the(________________(equation(and(then(we(________________(the(system.(

$

Ex.(2(Multiply(one(equation(to(have(opposite(coefficients,(then(rewrite(the(system(

1.2x + 3y = 6x + 2y = 5

((( ( ( 2.(4s − t = 95s + 2t = 8

((( ( 3.(6x − 4y = −85s + 2t = 8 $ $

$

$

$

After(reGwriting(the(system(we(will(always(____________(the(two(systems(together.(We(

____________(because(we(have(opposite(_________________.(When(we(add(one(of(the(

variables(will(be(___________________________.(((

We(then(solve(for(the(remaining(variable.$ After(we(have(solved(for(one(variable(we(

________________(that(value(into(either(_________________(equation(to(solve(for(the(other.(

We(then(write(our(solution(as(an(_____________________(pair.(

$(

Ex.(4(Solve(the(systems(

1.2x + 3y = 6x + 2y = 5

((( ( ( 2.(4s − t = 95s + 2t = 8

((( ( 3.(6x − 4y = −85s + 2t = 8 $ $

$$$$$$$$Remember:$

Step$1:$____________(the(equations(so(_________________(are(lined(up.(((

Step$2:$( ( ( (one(or(both(of(the(equations(by(a(number(to(obtain(

coefficients(that(are(opposites(for(one(of(the( ( ( .(((

Step$3:$$_________(the(equations(to(_______________(one(of(the((______________.$(

Step$4:$____________(the(equation(for(the(remaining(variable.(((

Step$5:$$$ $ $ $the(value(obtained(in(Step(4(into(either(one(of(the(

original(equations(and(solve(for(the(other( ( ( .$

Step$6:$$( ( (the(solution(as(an(_________________________________.(((

Solve:(

4a − 3b = −82a + 2b = 3

((

(

(

(

(

(

Don’t(forget!(It(is(possible(for(a(system(to(have(_____________,(_______________(or(

________________________(solutions.(If(each(variable(equals(a(number,(there(is(( (

( ( .(If(a(number(equals(itself,(there(are(! ! ! !

! .(If(a(number(equals(a(different(number,(there(is(! ! ! ! .(

Ex.(5(Solve(the(systems(

(

1.(2m + 3n = 4−m + 2n = 5

(( ( 2.(2x + 2y = 54x + 4y = 10

(( ( 3.(3x + 3y = 1221x + 21y = 10

$ $

$ ((($$$$$$Example$6.$A(buffet(has(one(price(for(adults(and(another(price(for(children.((The(Taylor(family(has(2(adults(and(3(children(and(their(bill(was($40.50.((The(Wong(family(has(3(adults(and(1(child(and(their(bill(was($38.((What(is(the(price(for(adults(and(children(at(the(buffet?((($$$$$$$Example$7(The(senior(classes(at(High(School(A(and(High(School(B(planned(separate(trips(to(NYC.(The(senior(class(at(High(School(A(rented(and(filled(1(van(and(6(buses(with(372(students.(High(School(B(rented(and(filled(4(vans(and(12(buses(with(780(students.(Each(van(and(each(bus(carried(the(same(number(of(students.(How(many(student(can(each(carry?($$$(

6"4$Elimination$Day$2$In$Class!! ! Name:_________________________________!! ! ! ! ! ! Date:___________________!Class:________!!Decide!which!equation!to!multiply!and!what!to!multiply!it!by:!

1.!!!!!3x − 4y = −4x + 3y = −10

! ! ! ! ! 2.!!!!4c − 3d = 222c − d = 10

!

!!Multiply!one!equation!to!have!opposite!coefficients,!then!rewrite!the!system

!

Solve!the!system!by!elimination!!

4.!!!!!3x − 4y = −4x + 3y = −10

! ! ! ! ! 5.!!!!4c − 3d = 222c − d = 10

!

!!!!Solve!the!following!systems:!!

6.!!!!!3x − 4y = −4x + 3y = −10

! ! ! ! ! 7.!!!!4c − 3d = 222c − d = 10

!

!!!!!

8.!!!!!6x − 4y = −84x + 2y = −3

! ! ! ! ! 9.!!!!4t − 6s = 122t − 3s = 6

!

!!!!!!!10.!!Brenda’s!school!is!selling!tickets!to!a!spring!musical.!On!the!first!day!of!ticket!sales!the!school!sold!3!senior!citizen!tickets!and!1!child!tickets!for!a!total!of!$38.!The!school!took!in!$52!on!the!second!day!by!selling!3!senior!citizen!tickets!and!2!child!tickets.!Find!the!price!of!both!tickets.!!!

6"5$:!Graphing$Linear$Inequalities$

Graphing!a!linear!inequality!is!almost!the!same!as!graphing!a!_________________!equation.!

Linear!inequalities!come!in!the!form!y!is!<,!>,!≤,≥ !!_______!+!________.!The!solution!to!a!linear!

inequality!is!a!__________________!instead!of!just!a!line.!

The!biggest!difference!for!a!linear!inequality!is!the!appearance!of!the!graph.!!

The!Line:!When!graphing!the!line!of!an!inequality:!!

If!the!inequality!has!a!≤!or!≥!use!a!!! ! ! .!

If!the!inequality!has!a!<!or!>!use!a!! ! ! ! .!

Ex.!1!Determine!if!the!line!is!dotted/!solid!!

1.! y > 2x − 9 !! ! 2.! y < 5x +12 ! ! 3.! y ≥ −3x −1 ! ! 4.! y ≤ 7x − 3.4 !

!

The!Shading:!When!graphing!a!linear!inequality!we!need!to!shade!our!graph!to!represent!

our!_________________!set.!

If!the!inequality!has!a!≤!or!<!shade!! ! ! ! .!

If!the!inequality!has!a!≥!or!>!shade!! ! ! ! .!

!

Ex.!2!Determine!if!we!shade!above!or!below!the!line!!

1.! y > 2x − 9 !! ! 2.! y < 5x +12 ! ! 3.! y ≥ −3x −1 ! ! 4.! y ≤ 7x − 3.4 !

!

We!graph!a!linear!inequality!the!same!way!we!graph!a!linear!equation.!First!we!____________!

the!_________Ointercept,!which!is!our!b!value.!We!then!use!our!________________(m)!to!plot!a!

second!point!on!our!line.!!Then!we!make!our!line!and!shading!according!to!our!

_____________________!sign.!

Ex.!3!Graph!the!following!Linear!Inequalities!!

1.! y ≥ 2x − 3 !! ! ! ! ! ! 2.! y < −3x + 2 !!

Line:! ! ! ! ! ! ! Line:!

Shading:! ! ! ! ! ! Shading:!

!

!!

3.! y ≤ 4x − 5 !! ! ! ! ! ! 4.! y > − 23x + 5 !!

Line:! ! ! ! ! ! ! Line:!

Shading:! ! ! ! ! ! Shading:!

!

!

!

!

!

!

! !

! !

!

!

! !

Horizontal$&$Vertical$Lines!

Remember,!Horizontal!lines!have!a!slope!of!____________!and!are!______=!equations.!Vertical!

lines!have!a!slope!that!is!________________________!and!are!________=!equations.!

When!shading!for!a!vertical!line!≥,> !!mean!to!shade!___________________!and!≤,< !mean!to!

shade!_________________.!For!a!horizontal!line≥,> !!mean!to!shade!___________________!and!≤,< !

mean!to!shade!_________________.!

1.! y ≤ 4 !! ! ! ! ! ! 2.! x > −2 !!

Line:! ! ! ! ! ! ! Line:!

Shading:! ! ! ! ! ! Shading:!

!

!

!

!

!

!

! !

! !

!

Application$Problem:$Robert!makes!$3.50!per!hour!working!at!a!convenience!store.!If!he!gets!a!bonus!of!$25!this!week,!how!many!hours!must!he!work!to!make!at!least!$165?!

a. Write!an!inequality!to!describe!this!situation!

!

!

!

b. Graph!to!find!the!solution!set.!

!

6"5$Graphing$Linear$Inequalities!! Name:_________________________________!IN!Class! ! ! ! ! Date:___________________!Class:________!!Determine!if!the!line!is!dotted/!solid!!

1.!!!!! y > 2x − 4 ! ! ! ! ! 2.!!!! y ≤ − 43x − 20 !

!Determine!if!we!shade!above!or!below!the!line!!

3.!!!!! y ≥ − 56x −18 ! ! ! ! ! 4.!!!! y > 3x + 9 !

!Graph!the!following!Linear!Inequalities!!

5.! y ≥ 2x − 3 !! ! ! ! 6.! y < −3x + 2 !!! ! ! 7.! y ≤ 4x − 5 !Line:! ! ! ! ! Line:! ! ! ! ! Line:!Shading:! ! ! ! Shading:! ! ! ! Shading:!

! !

!

8.! y > − 23x + 5 !! ! ! 9.! y ≥ −2 !! ! ! ! 10.! x < 5 !

Line:! ! ! ! ! Line:! ! ! ! ! Line:!Shading:! ! ! ! Shading:! ! ! ! Shading:!!

!!

6"6#:!Solving#Systems#of#Linear#Inequalities#

System#of#Linear#Inequalities#(System#of#Inequalities)#"!! ! ! ! !

! ! ! ! ! ! ! ! ! ! ! ! ! !

Solution#of#a#System!of#Inequalities#"!! ! ! ! ! ! ! !

! ! ! ! ! ! ! ! ! ! ! ! ! !

To!solve!a!system!of!inequalities,!we!are!going!solve!by!graphing,!like!we!did!on!the!first!

day.!The!major!difference!is!instead!of!having!_________!solution!we!will!have!a!_______________!

of!solutions.!We!will!graph!much!like!we!learned!how!to!last!class!period.!!

Remember:!

The!Line:!When!graphing!the!line!of!an!inequality:!!

If!the!inequality!has!a!≤!or!≥!use!a!!! ! ! .!

If!the!inequality!has!a!<!or!>!use!a!! ! ! ! .!

The!Shading:!When!graphing!a!linear!inequality!we!need!to!shade!our!graph!to!represent!

our!_________________!set.!

If!the!inequality!has!a!≤!or!<!shade!! ! ! ! .!

If!the!inequality!has!a!≥!or!>!shade!! ! ! ! .!

Ex.!1!Determine!in!the!following!systems!the!kind!of!line/shading!each!line!has:!

1.!y ≥ 2x − 3

y < 13x + 2

!! ! ! ! ! 2.!y > −4x +1

y ≤ 34x − 3

! ! !

!

3.!

y ≥ −3x + 3

y > 53x −1

x < 0

!!! ! ! ! 4.!

y ≥ 0

y ≤ 65x +1

x > 4

! ! !

!

To!find!the!solution!to!a!system!of!inequalities!we!want!to!look!for!where!the!________________!

of!each!line!______________________.!The!solution!will!be!where!____________!shadings!overlap.!

Remember!the!______________!to!an!inequality!is!a!______________________.!!

It!is!possible!for!a!system!of!linear!inequalities!to!have!___________!solution.!This!means!that!

the!______________________!does!not!overlap!at!all.!

To!solve!a!system!of!Inequalities:!

1.!_____________!each!equation!is!________"_______________!form.!

2.!______________!each!equation!on!the!coordinate!plane.!

If!the!inequality!has!a!≤!or!≥!use!a!!! ! ! .!

If!the!inequality!has!a!<!or!>!use!a!! ! ! ! .!

3.!________________!the!graph!according!to!the!inequalities!

If!the!inequality!has!a!≤!or!<!shade!! ! ! ! .!

If!the!inequality!has!a!≥!or!>!shade!! ! ! ! .!

4.!The!_________________!to!the!system!is!where!the!_______________!overlaps.!

Ex.!2!Solve!the!following!systems!by!graphing:!

1.!y ≤ −x − 2y ≥ −5x + 2

!! ! ! ! ! 2.!y < 3y ≤ −x +1

!

!

!

! !

!!

!

! ! !

!!

!

!

3.

y ≥ x − 3y > −x −1x < 3

!! ! ! ! ! 4.!y ≤ −x − 3y > −x + 3

#

#

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#

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#

#

Application#problems:!

The!theater!club!is!selling!shirts.!They!have!only!enough!supplies!to!print!120!shirts.!They!will!sell!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.!!

!

B.!Graph!the!system:!!

C.!Name!one!possible!solution.!

!

!

D.!Is!(45,!30)!a!solution?!

6"6#Graphing#Systems#of#Linear#Inequalities!! Name:_________________________________!IN!Class! ! ! ! ! ! Date:___________________!Class:________!!Ex.!1!Determine!in!the!following!systems!the!kind!of!line/shading!each!line!has:!

1.!y ≤ 2x − 3

y > 13x + 2

!! ! ! ! ! 2.!y < −4x +1

y > 34x − 3

! ! !

!

3.!

y < −3x + 3

y ≥ 53x −1

x ≤ 0

!!! ! ! ! 4.!

y < 0

y > 65x +1

x ≤ 4

! ! !

!Graph!the!following!Systems!of!Linear!Inequalities!!

5.! y ≤12x + 2

y < −2x − 3!!! ! ! 6.!

x ≤ −3

y < 53x + 2

!! ! ! ! 7.!y ≤ − 5

2− 2

y < − 12x + 2

!

! !

!!!!!!!!!!

8.!y ≥ 2

3x + 3

y > − 43x − 3

!! ! ! 9.!y < −2y > 4

!! ! ! ! 10.! x < 5y ≥ −x + 6

!

!