Hauv "system of equations", koj raug hais kom daws ob lossis ntau qhov sib npaug ib txhij. Thaum ob qhov sib npaug muaj ob qhov sib txawv sib txawv, piv txwv li x thiab y, kev daws teeb meem yuav zoo li nyuaj thaum xub thawj. Luckily, thaum koj paub tias koj yuav tsum ua dab tsi, koj tuaj yeem siv yooj yim siv koj cov txuj ci algebraic (thiab kev tshawb fawb txog kev suav cov feem) los daws qhov teeb meem. Kuj kawm paub kos ob qhov sib npaug no yog tias koj yog tus kawm paub pom, lossis xav tau los ntawm tus kws qhia ntawv. Cov duab kos yuav pab koj txheeb xyuas cov ntsiab lus lossis txheeb xyuas qhov txiaj ntsig ntawm koj txoj haujlwm. Txawm li cas los xij, txoj hauv kev no qeeb dua li lwm txoj hauv kev, thiab tsis tuaj yeem siv rau txhua qhov kab ke ntawm kab zauv.
Kauj ruam
Txoj Kev 1 ntawm 3: Siv Txoj Kev Hloov
Kauj Ruam 1. Txav mus rau qhov hloov pauv mus rau qhov tsis sib thooj ntawm qhov sib npaug
Txoj kev hloov pauv pib los ntawm "nrhiav tus nqi ntawm x" (lossis lwm yam sib txawv) hauv ib qho ntawm qhov sib npaug. Piv txwv li, hais qhov sib npaug ntawm qhov teeb meem yog 4x + 2 y = 8 thiab 5x + 3 y = 9. Pib los ntawm kev ua haujlwm ntawm thawj kab zauv. Rearrange qhov sib npaug los ntawm kev rho tawm 2y ntawm ob sab. Yog li, koj tau txais 4x = 8 - 2 y.
Txoj kev no feem ntau siv cov feem ntawm qhov kawg. Yog tias koj tsis nyiam suav cov feem, sim ua txoj hauv kev tshem tawm hauv qab no
Kauj Ruam 2. Faib ob sab ntawm qhov sib npaug kom "nrhiav tus nqi x"
Thaum lub sij hawm x (lossis yam sib txawv uas koj tab tom siv) nyob ib leeg ntawm ib sab ntawm qhov sib npaug, faib ob sab ntawm qhov kev ua zauv los ntawm qhov sib npaug kom tsuas yog qhov sib txawv tseem nyob. Piv txwv li:
- 4x = 8 - 2 y
- (4x)/4 = (8/4) - (2y/4)
- x = 2 - y
Kauj Ruam 3. Plug tus nqi x los ntawm thawj kab zauv mus rau qhov kev ua zauv ob
Nco ntsoov tias koj ntsaws nws rau qhov sib npaug thib ob, tsis yog qhov koj nyuam qhuav ua haujlwm. Hloov (hloov) qhov sib txawv x hauv qhov sib npaug thib ob. Yog li, qhov sib npaug thib ob tam sim no tsuas muaj ib qho sib txawv. Piv txwv li:
- Paub x = 2 - y.
- Koj qhov sib npaug thib ob yog 5x + 3 y = 9.
- Tom qab pauv tus x sib txawv hauv qhov sib npaug thib ob nrog tus nqi x los ntawm thawj qhov sib npaug, peb tau txais "2 - y": 5 (2 - y) + 3y = 9.
Kauj Ruam 4. Txhim kho qhov sib txawv uas tseem tshuav
Tam sim no, koj qhov sib npaug tsuas muaj ib qho sib txawv. Xam qhov sib npaug nrog kev ua lej algebraic zoo ib yam los nrhiav tus nqi ntawm qhov sib txawv. Yog tias ob qhov kev hloov pauv tshem tawm ib leeg, hla ncaj mus rau theem kawg. Txwv tsis pub koj yuav tau txais tus nqi rau ib qho ntawm qhov hloov pauv:
- 5 (2 - y) + 3y = 9
- 10 - (5/2) y + 3y = 9
- 10 - (5/2) y + (6/2) y = 9 (Yog tias koj tsis nkag siab cov kauj ruam no, kawm yuav ua li cas ntxiv cov zauv.)
- 10 + y = 9
- y = -1
- y = -2
Kauj Ruam 5. Siv cov lus teb tau los nrhiav qhov tseeb ntawm x hauv thawj qhov kev ua zauv
Tsis txhob tso tseg tam sim no vim tias koj lub laij lej tseem tsis tau tiav. Koj yuav tsum ntsaws qhov tau txais cov lus teb rau thawj qhov sib npaug kom pom tus nqi ntawm qhov sib txawv uas tseem tshuav:
- Paub y = -2
- Ib qho ntawm qhov sib npaug hauv thawj kab zauv yog 4x + 2 y = 8. (Koj tuaj yeem siv ib qho.)
- Hloov qhov y sib txawv nrog -2: 4x + 2 (-2) = 8.
- 4x - x = 8
- 4x = x 12
- x = 3 os
Kauj Ruam 6. Paub yuav ua li cas yog tias ob qho kev hloov pauv ib leeg tawm ib leeg
Thaum koj nkag mus x = 3y+2 lossis cov lus teb zoo ib yam rau qhov sib npaug thib ob, txhais tau tias koj tab tom sim kom tau qhov sib npaug uas tsuas muaj ib qho sib txawv. Qee zaum, koj tsuas yog tau txais qhov sib npaug tsis muaj sib txawv Muab ob npaug rau tshuaj xyuas koj cov haujlwm, thiab xyuas kom tseeb tias koj tau muab (rov ua dua tshiab) ib qho rau hauv kab zauv ob, tsis txhob rov qab mus rau thawj qhov sib npaug. Thaum koj paub tseeb tias koj tsis tau ua dab tsi tsis raug, sau ib qho ntawm cov hauv qab no:
- Yog tias qhov sib npaug tsis muaj qhov hloov pauv thiab tsis muaj tseeb (piv txwv, 3 = 5), qhov teeb meem no tsis muaj lus teb. (Thaum qhov no yog graphed, ob qhov kev ua zauv no zoo ib yam thiab tsis tau ntsib dua.)
- Yog hais tias qhov sib npaug tsis muaj qhov sib txawv thiab Yog lawm, (piv txwv 3 = 3), lub ntsiab lus uas cov lus nug muaj cov lus teb tsis txwv. Ib qho zauv ib yog raws nraim tib yam li kab zauv ob. (Thaum graphed, ob qhov sib npaug no yog tib txoj kab.)
Txoj Kev 2 ntawm 3: Siv Txoj Kev Raug Tshem Tawm
Kauj Ruam 1. Nrhiav qhov sib txawv ntawm qhov sib txawv
Qee zaum, qhov sib npaug hauv qhov teeb meem twb muaj lawm rho ib leeg thaum ntxiv. Piv txwv li, yog tias koj ua qhov sib npaug 3x + 2y = 11 thiab 5x - 4x = 13, cov ntsiab lus "+2y" thiab "-2y" yuav thim ib leeg tawm thiab tshem tawm qhov sib txawv "y" los ntawm kab zauv. Saib cov kab zauv hauv qhov teeb meem, thiab pom tias muaj qhov sib txawv uas tshem tawm ib leeg tawm, ib yam li hauv qhov piv txwv. Yog tsis yog, txuas ntxiv mus rau kauj ruam tom ntej.
Kauj Ruam 2. Muab cov zauv sib npaug los ntawm ib qho kom tshem tawm qhov sib txawv
(Hla cov kauj ruam no yog tias cov kev hloov pauv tau tso tseg ib leeg tawm.) Ua tibzoo saib cov piv txwv hauv qab no kom koj nkag siab lawv yooj yim:
- Qhov sib npaug hauv qhov teeb meem yog 3 x - 4x = 3 thiab - x + 2y = 4.
- Cia peb hloov thawj qhov sib npaug kom qhov sib txawv y rho tawm ib leeg tawm. (Koj tuaj yeem siv qhov sib txawv x kev. Cov lus teb zaum kawg tau yuav zoo ib yam.)
- Txawv txawv - ib y nyob rau hauv thawj kab zauv yuav tsum tau tshem tawm los ntawm + 2 xyoo hauv qhov kev ua zauv ob. Yuav ua li cas, sib npaug - ib y nrog 2.
- Muab ob tog ntawm qhov kev ua zauv faib ua 2, raws li hauv qab no: 2 (3x - y) = 2 (3), yog li 6x - 2y = 6. Tam sim no, pab pawg neeg - 2 xyi yuav rho tawm ib leeg tawm nrog +2 xyoo hauv qhov kev ua zauv ob.
Kauj Ruam 3. Muab ob qhov sib npaug
Qhov ua kom yuam kev yog txhawm rau ntxiv sab xis ntawm thawj kab zauv mus rau sab xis ntawm qhov sib npaug thib ob, thiab ntxiv sab laug ntawm thawj kab zauv mus rau sab laug ntawm qhov sib npaug thib ob. Yog ua tiav kom raug, ib qho ntawm qhov sib txawv yuav thim ib leeg tawm. Cia peb sim ua qhov kev xam los ntawm qhov piv txwv yav dhau los:
- Koj ob qhov sib npaug yog 6x - 2y = 6 thiab - x + 2y = 4.
- Ntxiv rau sab laug ntawm ob qhov sib npaug: 6x - 2y - x + 2y =?
- Ntxiv rau sab xis ntawm ob qhov sib npaug: 6x - 2y - x + 2y = 6 + 4.
Kauj Ruam 4. Tau txais tus nqi sib txawv kawg
Ua kom yooj yim rau koj qhov kev ua zauv sib xyaw, thiab ua haujlwm nrog tus lej algebra kom tau txais tus nqi ntawm qhov sib txawv kawg. Yog tias, tom qab ua kom yooj yim, qhov sib npaug tsis muaj qhov hloov pauv, txuas ntxiv mus rau theem kawg hauv ntu no.
Txwv tsis pub, koj yuav tau txais tus nqi rau ib qho ntawm cov kev hloov pauv. Piv txwv li:
- Paub 6x - 2y - x + 2y = 6 + 4.
- Pawg sib txawv x kev thiab y ua ke: 6x - x - 2y + 2y = 6 + 4.
- Ua kom yooj yim qhov sib npaug: 5 x = 4x 10
- Nrhiav tus nqi x: (5x)/5 = 10/5, kom tau x = 2 os.
Kauj Ruam 5. Nrhiav tus nqi ntawm lwm qhov sib txawv
Koj tau pom tus nqi ntawm ib qho sib txawv, tab sis dab tsi txog lwm qhov? Plug koj cov lus teb rau hauv ib qho ntawm qhov sib npaug kom pom tus nqi ntawm qhov sib txawv ntxiv. Piv txwv li:
- Paub x = 2 os, thiab ib qho ntawm qhov sib npaug hauv qhov teeb meem yog 3 x - 4x = 3.
- Hloov qhov x sib txawv nrog 2: 3 (2) - y = 3.
- Nrhiav tus nqi y hauv kab zauv: 6 - y = 3
- 6 - y + y = 3 + y, yog li 6 = 3 + y
- 3 = xyi
Kauj Ruam 6. Paub yuav ua dab tsi thaum ob qhov kev hloov pauv ib leeg tawm ib leeg
Qee zaum, ua ke ob qhov sib npaug ua rau muaj qhov sib npaug uas tsis nkag siab, lossis tsis pab koj daws qhov teeb meem. Txheeb xyuas koj txoj haujlwm, thiab yog tias koj paub tseeb tias koj tsis tau ua txhaum dab tsi, sau ib qho ntawm ob qho lus teb hauv qab no:
- Yog tias qhov sib npaug ua ke tsis muaj qhov hloov pauv thiab tsis muaj tseeb (piv txwv, 2 = 7), qhov teeb meem no tsis muaj lus teb. Cov lus teb no siv rau ob qho kev ua zauv. (Thaum qhov no yog graphed, ob qhov kev ua zauv no zoo ib yam thiab tsis tau ntsib dua.)
- Yog tias qhov sib npaug ua ke tsis muaj qhov sib txawv thiab Yog lawm, (piv txwv 0 = 0), lub ntsiab lus uas cov lus nug muaj cov lus teb tsis txwv. Ob qhov kev ua zauv no zoo ib yam rau ib leeg. (Thaum graphed, ob qhov sib npaug no yog kab ib yam.)
Txoj Kev 3 ntawm 3: Kos Kab Duab ntawm Kev Sib Piv
Kauj Ruam 1. Ua tus txheej txheem no tsuas yog thaum qhia
Tshwj tsis yog tias koj siv lub khoos phis tawj lossis lub tshuab xam zauv, txoj hauv kev no tsuas tuaj yeem muab cov lus teb kwv yees xwb. Koj tus kws qhia ntawv lossis phau ntawv yuav qhia koj kom siv txoj hauv kev no kom nkag mus rau tus cwj pwm ntawm kev kos kab zauv raws kab. Txoj kev no tseem tuaj yeem siv los tshuaj xyuas cov lus teb rau ib qho ntawm cov txheej txheem saum toj no.
Lub tswv yim tseem ceeb yog tias koj yuav tsum tau piav qhia ob qhov sib npaug thiab pom lawv cov ntsiab lus ntawm kev sib tshuam. Tus nqi x thiab y ntawm qhov chaw sib tshuam no yog lo lus teb rau qhov teeb meem
Kauj Ruam 2. Nrhiav y-qhov tseem ceeb ntawm ob qho kev ua zauv
Tsis txhob muab ob qhov sib npaug, thiab hloov txhua qhov sib npaug kom cov hom ntawv yog "y = _x + _". Piv txwv li:
- Koj thawj qhov sib npaug yog 2 x + 4x = 5. Hloov mus rau y = -2x + 5.
- Koj thawj qhov sib npaug yog - 3x + 6y = 0. Hloov mus rau 6 y = 4x + 3, thiab ua kom yooj yim rau y = x + 0.
- Yog tias koj ob qhov sib npaug yog tib yam nkaus, tag nrho kab yog "kev sib tshuam" ntawm ob qhov sib npaug. Sau cov lus teb tsis txwv raws li cov lus teb.
Kauj Ruam 3. Kos cov kab sib dhos
Kos kab ntsug "y-axis" kab thiab kab rov tav "x-axis" kab ntawm kab ntawv. Pib ntawm qhov chaw uas ob txoj kab sib tshuam sib luag (0, 0), sau cov lej cim 1, 2, 3, 4, thiab txuas ntxiv mus rau ntawm y-axis, thiab taw rau sab xis ntawm x-axis. Tom qab ntawd, sau tus lej cim -1, -2, thiab txuas ntxiv ua ntu ntu ntawm y -axis, thiab taw mus rau sab laug ntawm x -axis.
- Yog tias koj tsis muaj daim ntawv kos duab, siv tus pas ntsuas kom paub tseeb tias qhov sib nrug ntawm txhua tus lej yog tib yam nkaus.
- Yog tias koj siv tus lej loj lossis cov zauv, peb pom zoo kom ntsuas koj cov duab (piv txwv li 10, 20, 30 lossis 0, 1, 0, 2, 0, 3 siv 1, 2, 3).
Kauj Ruam 4. Kos tus y-cuam tshuam rau txhua qhov sib npaug
Yog hais tias qhov sib npaug nyob hauv daim ntawv y = _x + _, koj tuaj yeem pib kos duab los ntawm kev ua lub ntsiab lus uas kab kev sib npaug sib tshuam nrog y-axis. Tus nqi y yog ib txwm zoo ib yam li tus lej kawg hauv kab zauv.
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Txuas ntxiv qhov piv txwv dhau los, thawj kab (y = -2x + 5) sib tshuam y-axis ntawm
Kauj ruam 5.. kab thib ob (y = x + 0) sib tshuam y-axis ntawm 0. (Cov ntsiab lus no tau sau ua (0, 5) thiab (0, 0) ntawm kab ntawv.)
- Yog tias ua tau, kos kab thawj thiab kab thib ob nrog cwj mem xim sib txawv los yog xaum xim.
Kauj Ruam 5. Siv txoj kab nqes los txuas kab
Hauv hom sib npaug y = _x + _, tus lej nyob rau sab xub ntiag ntawm x qhia txog kab "theem nqes hav". Txhua lub sijhawm x tau nce los ntawm ib qho, tus nqi ntawm y yuav nce los ntawm tus lej ntawm theem nqes hav. Siv cov ntaub ntawv no txhawm rau nrhiav cov ntsiab lus rau txhua kab ntawm kab ntawv thaum x = 1. (Koj kuj tuaj yeem sau x = 1 hauv txhua qhov kev ua zauv thiab nrhiav tus nqi ntawm y.)
- Txuas ntxiv qhov piv txwv dhau los, kab y = -2x + 5 muaj qhov nqes hav - 2. Ntawm taw tes x = 1, kab txav mus nqis los ntawm 2 los ntawm qhov taw qhia x = 0. Kos kab txuas (0, 5) nrog (1, 3).
- Kab y = x + 0 muaj qhov nqes hav ½. Ntawm x = 1, kab txav mus caij los ntawm taw tes x = 0. Kos ib kab txuas (0, 0) nrog (1,).
- Yog ob kab muaj tib txoj kab nqes hav, ob leeg yuav tsis sib tshuam. Yog li no, kab ke ntawm kab zauv no tsis muaj lus teb. Sau tsis teb raws li cov lus teb.
Kauj Ruam 6. Txuas txuas cov kab kom txog thaum ob kab sib tshuam
Nres haujlwm thiab saib koj daim duab. yog tias ob kab tau hla ib leeg, txuas ntxiv mus rau kauj ruam tom ntej. Yog tsis yog, txiav txim siab raws li txoj haujlwm ntawm koj ob kab:
- Yog tias ob txoj kab mus rau ib leeg, txuas ntxiv txuas cov kab ntawm koj kab txaij.
- Yog tias ob kab txav deb ntawm ib leeg, rov qab thiab txuas cov dots hauv cov lus qhia rov qab, pib ntawm x = 1.
- Yog tias ob kab sib nrug deb heev, sim dhia hla thiab txuas cov ntsiab lus kom deb dua, piv txwv li x = 10.
Kauj Ruam 7. Nrhiav cov lus teb ntawm qhov chaw sib tshuam
Tom qab ob kab sib tshuam, tus nqi ntawm x thiab y ntawm qhov ntawd yog lo lus teb rau koj qhov teeb meem. Yog tias koj muaj hmoo, cov lus teb yuav suav tag nrho. Piv txwv li, hauv peb qhov piv txwv ob kab sib tshuam ntawm lub ntsiab lus (2, 1) yog li cov lus teb yog x = 2 thiab y = 1. Hauv qee qhov kab ke ntawm kab zauv, cov ntsiab lus uas kab sib tshuam yog nruab nrab ntawm ob tus lej, thiab yog tias kab ntawv tsis raug, nws nyuaj rau taw qhia qhov twg x thiab y qhov tseem ceeb nyob ntawm qhov chaw sib tshuam. Yog tias tau tso cai, koj tuaj yeem sau "x nyob nruab nrab ntawm 1 thiab 2" raws li cov lus teb, lossis siv qhov kev hloov lossis tshem tawm txoj hauv kev los nrhiav cov lus teb.
Lub tswv yim
- Koj tuaj yeem tshawb xyuas koj txoj haujlwm los ntawm kev ntsaws cov lus teb rau hauv qhov sib npaug qub. Yog tias qhov kev ua zauv dhau los ua qhov tseeb (piv txwv li 3 = 3), nws txhais tau tias koj cov lus teb raug lawm.
- Thaum siv txoj hauv kev tshem tawm, qee zaum koj yuav tsum muab cov zauv sib npaug los ntawm tus lej tsis zoo kom cov kev hloov pauv tuaj yeem thim tawm ib leeg.
