Whakaoti i te {l}{3x+10y=11}{-10x-8y=14}

$$\left. \begin{array} { l } { 3 x + 10 y = 11 } \\ { - 10 x - 8 y = 14 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-3$$

$$y=2$$

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.

$$3x+10y=11,-10x-8y=14$$

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

$$3x+10y=11$$

Me tango 10y mai i ngā taha e rua o te whārite.

$$3x=-10y+11$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(-10y+11\right)$$

Whakareatia \frac{1}{3} ki te -10y+11.

$$x=-\frac{10}{3}y+\frac{11}{3}$$

Whakakapia te \frac{-10y+11}{3} mō te x ki tērā atu whārite, -10x-8y=14.

$$-10\left(-\frac{10}{3}y+\frac{11}{3}\right)-8y=14$$

Whakareatia -10 ki te \frac{-10y+11}{3}.

$$\frac{100}{3}y-\frac{110}{3}-8y=14$$

Tāpiri \frac{100y}{3} ki te -8y.

$$\frac{76}{3}y-\frac{110}{3}=14$$

Me tāpiri \frac{110}{3} ki ngā taha e rua o te whārite.

$$\frac{76}{3}y=\frac{152}{3}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{76}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=2$$

Whakaurua te 2 mō y ki x=-\frac{10}{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.

$$x=-\frac{10}{3}\times 2+\frac{11}{3}$$

Whakareatia -\frac{10}{3} ki te 2.

$$x=\frac{-20+11}{3}$$

Tāpiri \frac{11}{3} ki te -\frac{20}{3} 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.

$$x=-3$$

Kua oti te pūnaha te whakatau.

$$x=-3,y=2$$

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.

$$3x+10y=11,-10x-8y=14$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&10\\-10&-8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}11\\14\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&10\\-10&-8\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&10\\-10&-8\end{matrix}\right))\left(\begin{matrix}3&10\\-10&-8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&10\\-10&-8\end{matrix}\right))\left(\begin{matrix}11\\14\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&10\\-10&-8\end{matrix}\right))\left(\begin{matrix}11\\14\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&10\\-10&-8\end{matrix}\right))\left(\begin{matrix}11\\14\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{8}{3\left(-8\right)-10\left(-10\right)}&-\frac{10}{3\left(-8\right)-10\left(-10\right)}\\-\frac{-10}{3\left(-8\right)-10\left(-10\right)}&\frac{3}{3\left(-8\right)-10\left(-10\right)}\end{matrix}\right)\left(\begin{matrix}11\\14\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{19}&-\frac{5}{38}\\\frac{5}{38}&\frac{3}{76}\end{matrix}\right)\left(\begin{matrix}11\\14\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{19}\times 11-\frac{5}{38}\times 14\\\frac{5}{38}\times 11+\frac{3}{76}\times 14\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-3\\2\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-3,y=2$$

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.

$$3x+10y=11,-10x-8y=14$$

Kia ōrite ai a 3x me -10x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -10 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$-10\times 3x-10\times 10y=-10\times 11,3\left(-10\right)x+3\left(-8\right)y=3\times 14$$

Whakarūnātia.

$$-30x-100y=-110,-30x-24y=42$$

Me tango -30x-24y=42 mai i -30x-100y=-110 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-30x+30x-100y+24y=-110-42$$

Tāpiri -30x ki te 30x. Ka whakakore atu ngā kupu -30x me 30x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-100y+24y=-110-42$$

Tāpiri -100y ki te 24y.

$$-76y=-110-42$$

Tāpiri -110 ki te -42.

$$-76y=-152$$

Whakawehea ngā taha e rua ki te -76.

$$y=2$$

Whakaurua te 2 mō y ki -10x-8y=14. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-10x-8\times 2=14$$

Whakareatia -8 ki te 2.

$$-10x-16=14$$

Me tāpiri 16 ki ngā taha e rua o te whārite.

$$-10x=30$$

Whakawehea ngā taha e rua ki te -10.

$$x=-3$$

Kua oti te pūnaha te whakatau.

$$x=-3,y=2$$