Whakaoti i te {l}{5x-6y=-25}{4x-3y+20=0}

$$\left. \begin{array} { l } { 5 x - 6 y = - 25 } \\ { 4 x - 3 y + 20 = 0 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-5$$

$$y=0$$

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.

$$5x-6y=-25,4x-3y+20=0$$

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

$$5x-6y=-25$$

Me tāpiri 6y ki ngā taha e rua o te whārite.

$$5x=6y-25$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{5}\left(6y-25\right)$$

Whakareatia \frac{1}{5} ki te 6y-25.

$$x=\frac{6}{5}y-5$$

Whakakapia te \frac{6y}{5}-5 mō te x ki tērā atu whārite, 4x-3y+20=0.

$$4\left(\frac{6}{5}y-5\right)-3y+20=0$$

Whakareatia 4 ki te \frac{6y}{5}-5.

$$\frac{24}{5}y-20-3y+20=0$$

Tāpiri \frac{24y}{5} ki te -3y.

$$\frac{9}{5}y-20+20=0$$

Tāpiri -20 ki te 20.

$$\frac{9}{5}y=0$$

Whakawehea ngā taha e rua o te whārite ki te \frac{9}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=0$$

Whakaurua te 0 mō y ki x=\frac{6}{5}y-5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-5$$

Kua oti te pūnaha te whakatau.

$$x=-5,y=0$$

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.

$$5x-6y=-25,4x-3y+20=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&-6\\4&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-25\\-20\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&-6\\4&-3\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&-6\\4&-3\end{matrix}\right))\left(\begin{matrix}5&-6\\4&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-6\\4&-3\end{matrix}\right))\left(\begin{matrix}-25\\-20\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-6\\4&-3\end{matrix}\right))\left(\begin{matrix}-25\\-20\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-6\\4&-3\end{matrix}\right))\left(\begin{matrix}-25\\-20\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{5\left(-3\right)-\left(-6\times 4\right)}&-\frac{-6}{5\left(-3\right)-\left(-6\times 4\right)}\\-\frac{4}{5\left(-3\right)-\left(-6\times 4\right)}&\frac{5}{5\left(-3\right)-\left(-6\times 4\right)}\end{matrix}\right)\left(\begin{matrix}-25\\-20\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{3}&\frac{2}{3}\\-\frac{4}{9}&\frac{5}{9}\end{matrix}\right)\left(\begin{matrix}-25\\-20\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{3}\left(-25\right)+\frac{2}{3}\left(-20\right)\\-\frac{4}{9}\left(-25\right)+\frac{5}{9}\left(-20\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-5\\0\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-5,y=0$$

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.

$$5x-6y=-25,4x-3y+20=0$$

Kia ōrite ai a 5x 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 5.

$$4\times 5x+4\left(-6\right)y=4\left(-25\right),5\times 4x+5\left(-3\right)y+5\times 20=0$$

Whakarūnātia.

$$20x-24y=-100,20x-15y+100=0$$

Me tango 20x-15y+100=0 mai i 20x-24y=-100 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$20x-20x-24y+15y-100=-100$$

Tāpiri 20x ki te -20x. Ka whakakore atu ngā kupu 20x me -20x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-24y+15y-100=-100$$

Tāpiri -24y ki te 15y.

$$-9y-100=-100$$

Me tāpiri 100 ki ngā taha e rua o te whārite.

$$-9y=0$$

Whakawehea ngā taha e rua ki te -9.

$$y=0$$

Whakaurua te 0 mō y ki 4x-3y+20=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x+20=0$$

Me tango 20 mai i ngā taha e rua o te whārite.

$$4x=-20$$

Whakawehea ngā taha e rua ki te 4.

$$x=-5$$

Kua oti te pūnaha te whakatau.

$$x=-5,y=0$$