Whakaoti i te {l}{3x+2y=11}{4x+9y=117}

$$\left. \begin{array} { l } { 3 x + 2 y = 11 } \\ { 4 x + 9 y = 117 } \end{array} \right.$$

Whakaoti mō x, y

$$x = -\frac{135}{19} = -7\frac{2}{19} \approx -7.105263158$$

$$y = \frac{307}{19} = 16\frac{3}{19} \approx 16.157894737$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Hei whakaoti i ētahi whārite takirua mā te whakakapinga, me whakaoti tētahi whārite i te tuatahi mō tētahi o ngā taurangi. Ka whakakapi i te otinga mō taua taurangi ki tērā o ngā whārite.

$$3x+2y=11,4x+9y=117$$

Kōwhiria tētahi o ngā whārite ka whakaotia mō te x mā te wehe i te x i te taha mauī o te tohu ōrite.

$$3x+2y=11$$

Me tango 2y mai i ngā taha e rua o te whārite.

$$3x=-2y+11$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(-2y+11\right)$$

Whakareatia \frac{1}{3} ki te -2y+11.

$$x=-\frac{2}{3}y+\frac{11}{3}$$

Whakakapia te \frac{-2y+11}{3} mō te x ki tērā atu whārite, 4x+9y=117.

$$4\left(-\frac{2}{3}y+\frac{11}{3}\right)+9y=117$$

Whakareatia 4 ki te \frac{-2y+11}{3}.

$$-\frac{8}{3}y+\frac{44}{3}+9y=117$$

Tāpiri -\frac{8y}{3} ki te 9y.

$$\frac{19}{3}y+\frac{44}{3}=117$$

Me tango \frac{44}{3} mai i ngā taha e rua o te whārite.

$$\frac{19}{3}y=\frac{307}{3}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{19}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{307}{19}$$

Whakaurua te \frac{307}{19} mō y ki x=-\frac{2}{3}y+\frac{11}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{2}{3}\times \frac{307}{19}+\frac{11}{3}$$

Whakareatia -\frac{2}{3} ki te \frac{307}{19} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=-\frac{614}{57}+\frac{11}{3}$$

Tāpiri \frac{11}{3} ki te -\frac{614}{57} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=-\frac{135}{19}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{135}{19},y=\frac{307}{19}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$3x+2y=11,4x+9y=117$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&2\\4&9\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}11\\117\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&2\\4&9\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&2\\4&9\end{matrix}\right))\left(\begin{matrix}3&2\\4&9\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&2\\4&9\end{matrix}\right))\left(\begin{matrix}11\\117\end{matrix}\right)$$

Ko te hua o tētahi poukapa me te kōaro ko te poukapa tuakiri.

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&2\\4&9\end{matrix}\right))\left(\begin{matrix}11\\117\end{matrix}\right)$$

Whakareatia ngā poukapa kei te taha mauī o te tohu ōrite.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&2\\4&9\end{matrix}\right))\left(\begin{matrix}11\\117\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right) te poukapa kōaro, nō reira ka taea te tuhi anō te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{9}{3\times 9-2\times 4}&-\frac{2}{3\times 9-2\times 4}\\-\frac{4}{3\times 9-2\times 4}&\frac{3}{3\times 9-2\times 4}\end{matrix}\right)\left(\begin{matrix}11\\117\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{9}{19}&-\frac{2}{19}\\-\frac{4}{19}&\frac{3}{19}\end{matrix}\right)\left(\begin{matrix}11\\117\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{9}{19}\times 11-\frac{2}{19}\times 117\\-\frac{4}{19}\times 11+\frac{3}{19}\times 117\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{135}{19}\\\frac{307}{19}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{135}{19},y=\frac{307}{19}$$

Ngā Upane e Whakamahi i te Tangohanga

Hei whakaoti mā te tangohanga, ko ngā tau whakarea o tētahi o ngā taurangi me mātua ōrite i ngā whārite e rua kia whakakorehia ai te taurangi ina tangohia tētahi whārite mai i tētahi atu.

$$3x+2y=11,4x+9y=117$$

Kia ōrite ai a 3x me 4x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 4 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$4\times 3x+4\times 2y=4\times 11,3\times 4x+3\times 9y=3\times 117$$

Whakarūnātia.

$$12x+8y=44,12x+27y=351$$

Me tango 12x+27y=351 mai i 12x+8y=44 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$12x-12x+8y-27y=44-351$$

Tāpiri 12x ki te -12x. Ka whakakore atu ngā kupu 12x me -12x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$8y-27y=44-351$$

Tāpiri 8y ki te -27y.

$$-19y=44-351$$

Tāpiri 44 ki te -351.

$$-19y=-307$$

Whakawehea ngā taha e rua ki te -19.

$$y=\frac{307}{19}$$

Whakaurua te \frac{307}{19} mō y ki 4x+9y=117. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x+9\times \frac{307}{19}=117$$

Whakareatia 9 ki te \frac{307}{19}.

$$4x+\frac{2763}{19}=117$$

Me tango \frac{2763}{19} mai i ngā taha e rua o te whārite.

$$4x=-\frac{540}{19}$$

Whakawehea ngā taha e rua ki te 4.

$$x=-\frac{135}{19}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{135}{19},y=\frac{307}{19}$$