Whakaoti i te {l}{10x+18y=-1}{16x-9y=-5}

$$\left\{ \begin{array} { l } { 10 x + 18 y = - 1 } \\ { 16 x - 9 y = - 5 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-\frac{11}{42}\approx -0.261904762$$

$$y=\frac{17}{189}\approx 0.08994709$$

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.

$$10x+18y=-1,16x-9y=-5$$

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

$$10x+18y=-1$$

Me tango 18y mai i ngā taha e rua o te whārite.

$$10x=-18y-1$$

Whakawehea ngā taha e rua ki te 10.

$$x=\frac{1}{10}\left(-18y-1\right)$$

Whakareatia \frac{1}{10} ki te -18y-1.

$$x=-\frac{9}{5}y-\frac{1}{10}$$

Whakakapia te -\frac{9y}{5}-\frac{1}{10} mō te x ki tērā atu whārite, 16x-9y=-5.

$$16\left(-\frac{9}{5}y-\frac{1}{10}\right)-9y=-5$$

Whakareatia 16 ki te -\frac{9y}{5}-\frac{1}{10}.

$$-\frac{144}{5}y-\frac{8}{5}-9y=-5$$

Tāpiri -\frac{144y}{5} ki te -9y.

$$-\frac{189}{5}y-\frac{8}{5}=-5$$

Me tāpiri \frac{8}{5} ki ngā taha e rua o te whārite.

$$-\frac{189}{5}y=-\frac{17}{5}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{189}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{17}{189}$$

Whakaurua te \frac{17}{189} mō y ki x=-\frac{9}{5}y-\frac{1}{10}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{9}{5}\times \frac{17}{189}-\frac{1}{10}$$

Whakareatia -\frac{9}{5} ki te \frac{17}{189} 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.

$$x=-\frac{17}{105}-\frac{1}{10}$$

Tāpiri -\frac{1}{10} ki te -\frac{17}{105} 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.

$$x=-\frac{11}{42}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{11}{42},y=\frac{17}{189}$$

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.

$$10x+18y=-1,16x-9y=-5$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}10&18\\16&-9\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-1\\-5\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}10&18\\16&-9\end{matrix}\right).

$$inverse(\left(\begin{matrix}10&18\\16&-9\end{matrix}\right))\left(\begin{matrix}10&18\\16&-9\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}10&18\\16&-9\end{matrix}\right))\left(\begin{matrix}-1\\-5\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}10&18\\16&-9\end{matrix}\right))\left(\begin{matrix}-1\\-5\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}10&18\\16&-9\end{matrix}\right))\left(\begin{matrix}-1\\-5\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{9}{10\left(-9\right)-18\times 16}&-\frac{18}{10\left(-9\right)-18\times 16}\\-\frac{16}{10\left(-9\right)-18\times 16}&\frac{10}{10\left(-9\right)-18\times 16}\end{matrix}\right)\left(\begin{matrix}-1\\-5\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{42}&\frac{1}{21}\\\frac{8}{189}&-\frac{5}{189}\end{matrix}\right)\left(\begin{matrix}-1\\-5\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{42}\left(-1\right)+\frac{1}{21}\left(-5\right)\\\frac{8}{189}\left(-1\right)-\frac{5}{189}\left(-5\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{11}{42}\\\frac{17}{189}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{11}{42},y=\frac{17}{189}$$

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.

$$10x+18y=-1,16x-9y=-5$$

Kia ōrite ai a 10x 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 10.

$$16\times 10x+16\times 18y=16\left(-1\right),10\times 16x+10\left(-9\right)y=10\left(-5\right)$$

Whakarūnātia.

$$160x+288y=-16,160x-90y=-50$$

Me tango 160x-90y=-50 mai i 160x+288y=-16 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$160x-160x+288y+90y=-16+50$$

Tāpiri 160x ki te -160x. Ka whakakore atu ngā kupu 160x me -160x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$288y+90y=-16+50$$

Tāpiri 288y ki te 90y.

$$378y=-16+50$$

Tāpiri -16 ki te 50.

$$378y=34$$

Whakawehea ngā taha e rua ki te 378.

$$y=\frac{17}{189}$$

Whakaurua te \frac{17}{189} mō y ki 16x-9y=-5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$16x-9\times \frac{17}{189}=-5$$

Whakareatia -9 ki te \frac{17}{189}.

$$16x-\frac{17}{21}=-5$$

Me tāpiri \frac{17}{21} ki ngā taha e rua o te whārite.

$$16x=-\frac{88}{21}$$

Whakawehea ngā taha e rua ki te 16.

$$x=-\frac{11}{42}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{11}{42},y=\frac{17}{189}$$