Lesson8
SolvingLinearEquations–MorePractice
3b=-10
b=-10/3
c/8=-3
c=-3×8
=-24
3t–(-4)=19
3t+4=19
3t=19–4
3t=15
t=5
-8h–44=-4
-8h=-4+44
-8h=40
h=-5
2t/5+10=20
2t/5=20–10
2t/5=10
2t=10×5
=50
t=25
-50/u–(-10)=60
6–4/f=0
Homework:Handouts
Lesson9
*Extension*Simultaneoushandoutreviewofgraphingusingatableofvalues
+graphingusingy=mx+b;preparestudentstoassistothersinsolving
simultaneousquestionsalgebraicallyafterthequiz
• Remember:Eachequationactuallydefinesaline
o Linearversusnon-linearalgebra
• Introducey=mx+b
§ m=slope=rise/run=howsteepalineis,andinwhichdirection
• 2x=2/1xà2rise/1run[plot]
• 1/2xà1rise/2run[plot]
• 5x=5/1xà5rise/1run[plot]
• -1/7xà-1rise/7run[plot]
§ y-intercept=wherethelinecrossesthey-axis(i.e.,they-valuewhenx=
0)
• y=2x–8à(0,-8)
• y=3x+4à(0,4)
• 2x–y=-5à2(0)–y=5ày=-5à(0,-5)
• x–4y=5à(0)–4y=5ày=5/4à(0,5/4)
§ Ploty-interceptandfromthere,drawouttheslopetocreatetheline
• Don’tforgettodrawarrowsonbothendsoftheline
• Drawbothequationsonthesamegraphandlabelthepointof
intersection
Howdothepointsofintersectionrelatetoyouralgebraicsolutions?
Lesson10
SimultaneousLinearEquations
Howcanmathematicalmodels(tables,graphs,equations)beusedtodisplay
anddescribearealworldsituation?
Linearequationsusevariablesthatdonothaveexponents,e.g.,y=3x–4.When
graphed,theyformstraightlines.Non-linearequationsusevariablesthatdohave
exponents,e.g.,y=2x2+7.Whengraphed,theyformcurves.
Simultaneouslinearequationsinvolvesolvingmultiplevariablesinmultiple
equations,allwithoutexponents.Theequationsaresolvedatthesametime.Each
variablehasthesamevalueineachequation.Forexample,
3x+2y=12
2x+y=7
arebothlinearequations.Theycanbesolvedsimultaneously.Forbothequations,
thesolutionisx=2andy=3.
3x+2y=12 à
3(2)+2(3)=12✓
2x+y=7
à
2(2)+3=7✓
Wewillbeusingthesubstitutionmethodtosolvesimultaneouslinearequations.
SolvingSimultaneousLinearEquationsUsingtheSubstitutionMethod(Part1)
Step1:SolveForOneVariable
2x+y=10
x+y=7
Solveforx: 2x+y=10 the“y”value
2x+7–x=10
2x–x=10–7
x=3
• 2(3)+y=10
• 6+4=10
Solveforx: 2x+y=10 y=10–2x…plugthisintothe“y”value
Solveforx:
y=7–x……now…plugthisinto
Solveforx:
x+y=7
x+10–2x=7
x–2x=7–10
-1x=7–10
-1x=-3
x=3
2x+y=10
x+y=7
Solvefory: 2x+y=10 2(7–y)+y=10
value
14–2y+y=10
-2y+y=10–14
-y=-4
y=4
•
•
Solvefory:
x+y=7
x=7–y…plugthisintothe“x”
2x+4=10
3+4=7
Solvefory: Solvefory:
2x+y=10 x+y=7
2x/2+y/2=10/2 5–0.5y+y=7
x+1y/2=10/2
-0.5y+y=7–5
x+0.5y=5 0.5y=2
x=5–0.5y…plugthisintothe“x”value y=2/0.5
y=4
Foreachpairofequations,solveforbothvariablesinbothequations:
2x–y=6
x+y=6
2x–y=6
x+y=6
Solveforx: Solveforx:
2x=6+y
x=6-y
x=(6+y)/2 =3+y/2
=3+1/2y
Solvefory: Solvefory:
2x–6=y
y=6-x
y=2x–6
Homework:TalkandTextCellPlanActivity
Quiz#1!
Lesson11
SolvingSimultaneousLinearEquationsUsingtheSubstitutionMethod(Part2)
Step1:SolveForOneVariable–selecttheeasiervariableandequationtosolvefor;if
possibleavoidfractions.
2x+3y=15
x–3y=-6
x=-6+3y
Step2:Substituteyoursolutionforthevariableintotheotherequation,thensolvefor
thevariable
x=-6+3yforx,into2x+3y=15
2(-6+3y)+3y=15
-12+6y+3y=15
-12+9y=15
9y=15+12
9y=27
y=3
Step3:SubstituteyoursolutionfromStep2intooneoftheoriginalequationsthen
solvefortheothervariable
y=3intox–3y=-6
x–3(3)=-6
x–9=-6
x=-6+9
=3
Step4:Checkyourworkandstateyouranswer
x=3,y=3
2x+3y=15à2(3)+3(3)=6+9=15
x–3y=-6à3–3(3)=3–9=-6
∴x=3,y=3
MoreExamples
Step1:SolveForOneVariable
3x+6=2y
3x+8y+21=0
(3x+6)/2=y
3/2x+3=y
Step2:Substituteyoursolutionforthevariableintotheotherequation,thensolvefor
thevariable
y=3/2x+3fory,into3x+8y+21=0
3x+8(3/2x+3)+21=0
3x+24/2x+24+21=0
3x+12x+24+21=0
15x=-21–24
15x=-45
x=-3
Step3:SubstituteyoursolutionfromStep2intooneoftheoriginalequationsthen
solvefortheothervariable
x=-3into3x+6=2y
3(-3)+6=2y
-9+6=2y
-3=2y
y=-3/2
Step4:Checkyourworkandstateyouranswer
x=-3,y=-3/2
3x+6=2yà3(-3)+6=2(-3/2)à-9+6=-3à-3=-3
3x+8y+21=0à3(-3)+8(-3/2)+21=0à-9+-24/2+21=0
à-9+-12+21=0à0=0
∴x=-3,y=-3/2
Homework:Handout–solveforbothvariables