Whakaoti i te {l}{b*12x-15y=-4}{16x+10y=7}

$$\left. \begin{array} { l } { b \cdot 12 x - 15 y = - 4 } \\ { 16 x + 10 y = 7 } \end{array} \right.$$

Whakaoti mō x, y

$$x=\frac{13}{24\left(b+2\right)}$$

$$y=\frac{21b+16}{30\left(b+2\right)}$$

$$b\neq -2$$

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.

$$12bx-15y=-4,16x+10y=7$$

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

$$12bx-15y=-4$$

Me tāpiri 15y ki ngā taha e rua o te whārite.

$$12bx=15y-4$$

Whakawehea ngā taha e rua ki te 12b.

$$x=\frac{1}{12b}\left(15y-4\right)$$

Whakareatia \frac{1}{12b} ki te 15y-4.

$$x=\frac{5}{4b}y-\frac{1}{3b}$$

Whakakapia te \frac{-4+15y}{12b} mō te x ki tērā atu whārite, 16x+10y=7.

$$16\left(\frac{5}{4b}y-\frac{1}{3b}\right)+10y=7$$

Whakareatia 16 ki te \frac{-4+15y}{12b}.

$$\frac{20}{b}y-\frac{16}{3b}+10y=7$$

Tāpiri \frac{20y}{b} ki te 10y.

$$\left(10+\frac{20}{b}\right)y-\frac{16}{3b}=7$$

Me tāpiri \frac{16}{3b} ki ngā taha e rua o te whārite.

$$\left(10+\frac{20}{b}\right)y=7+\frac{16}{3b}$$

Whakawehea ngā taha e rua ki te \frac{20}{b}+10.

$$y=\frac{21b+16}{30\left(b+2\right)}$$

Whakaurua te \frac{16+21b}{30\left(2+b\right)} mō y ki x=\frac{5}{4b}y-\frac{1}{3b}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{5}{4b}\times \frac{21b+16}{30\left(b+2\right)}-\frac{1}{3b}$$

Whakareatia \frac{5}{4b} ki te \frac{16+21b}{30\left(2+b\right)}.

$$x=\frac{21b+16}{24b\left(b+2\right)}-\frac{1}{3b}$$

Tāpiri -\frac{1}{3b} ki te \frac{16+21b}{24b\left(2+b\right)}.

$$x=\frac{13}{24\left(b+2\right)}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{13}{24\left(b+2\right)},y=\frac{21b+16}{30\left(b+2\right)}$$

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.

$$12bx-15y=-4,16x+10y=7$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}12b&-15\\16&10\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-4\\7\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}12b&-15\\16&10\end{matrix}\right).

$$inverse(\left(\begin{matrix}12b&-15\\16&10\end{matrix}\right))\left(\begin{matrix}12b&-15\\16&10\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}12b&-15\\16&10\end{matrix}\right))\left(\begin{matrix}-4\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}12b&-15\\16&10\end{matrix}\right))\left(\begin{matrix}-4\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}12b&-15\\16&10\end{matrix}\right))\left(\begin{matrix}-4\\7\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{10}{12b\times 10-\left(-15\times 16\right)}&-\frac{-15}{12b\times 10-\left(-15\times 16\right)}\\-\frac{16}{12b\times 10-\left(-15\times 16\right)}&\frac{12b}{12b\times 10-\left(-15\times 16\right)}\end{matrix}\right)\left(\begin{matrix}-4\\7\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{12\left(b+2\right)}&\frac{1}{8\left(b+2\right)}\\-\frac{2}{15\left(b+2\right)}&\frac{b}{10\left(b+2\right)}\end{matrix}\right)\left(\begin{matrix}-4\\7\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{12\left(b+2\right)}\left(-4\right)+\frac{1}{8\left(b+2\right)}\times 7\\\left(-\frac{2}{15\left(b+2\right)}\right)\left(-4\right)+\frac{b}{10\left(b+2\right)}\times 7\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{13}{24\left(b+2\right)}\\\frac{21b+16}{30\left(b+2\right)}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{13}{24\left(b+2\right)},y=\frac{21b+16}{30\left(b+2\right)}$$

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.

$$12bx-15y=-4,16x+10y=7$$

Kia ōrite ai a 12bx me 16x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 16 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 12b.

$$16\times 12bx+16\left(-15\right)y=16\left(-4\right),12b\times 16x+12b\times 10y=12b\times 7$$

Whakarūnātia.

$$192bx-240y=-64,192bx+120by=84b$$

Me tango 192bx+120by=84b mai i 192bx-240y=-64 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$192bx+\left(-192b\right)x-240y+\left(-120b\right)y=-64-84b$$

Tāpiri 192bx ki te -192bx. Ka whakakore atu ngā kupu 192bx me -192bx, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-240y+\left(-120b\right)y=-64-84b$$

Tāpiri -240y ki te -120by.

$$\left(-120b-240\right)y=-64-84b$$

Tāpiri -64 ki te -84b.

$$\left(-120b-240\right)y=-84b-64$$

Whakawehea ngā taha e rua ki te -240-120b.

$$y=\frac{21b+16}{30\left(b+2\right)}$$

Whakaurua te \frac{16+21b}{30\left(2+b\right)} mō y ki 16x+10y=7. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$16x+10\times \frac{21b+16}{30\left(b+2\right)}=7$$

Whakareatia 10 ki te \frac{16+21b}{30\left(2+b\right)}.

$$16x+\frac{21b+16}{3\left(b+2\right)}=7$$

Me tango \frac{16+21b}{3\left(2+b\right)} mai i ngā taha e rua o te whārite.

$$16x=\frac{26}{3\left(b+2\right)}$$

Whakawehea ngā taha e rua ki te 16.

$$x=\frac{13}{24\left(b+2\right)}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{13}{24\left(b+2\right)},y=\frac{21b+16}{30\left(b+2\right)}$$