Whakaoti i te {c}{x+y=89}{2x+y=23y}

$$\left. \begin{array} { c } { x + y = 89 } \\ { 2 x + y = 23 y } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{979}{12} = 81\frac{7}{12} \approx 81.583333333$$

$$y = \frac{89}{12} = 7\frac{5}{12} \approx 7.416666667$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Tangohia te 23y mai i ngā taha e rua.

$$2x+y-23y=0$$

Pahekotia te y me -23y, ka -22y.

$$2x-22y=0$$

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.

$$x+y=89,2x-22y=0$$

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

$$x+y=89$$

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

$$x=-y+89$$

Whakakapia te -y+89 mō te x ki tērā atu whārite, 2x-22y=0.

$$2\left(-y+89\right)-22y=0$$

Whakareatia 2 ki te -y+89.

$$-2y+178-22y=0$$

Tāpiri -2y ki te -22y.

$$-24y+178=0$$

Me tango 178 mai i ngā taha e rua o te whārite.

$$-24y=-178$$

Whakawehea ngā taha e rua ki te -24.

$$y=\frac{89}{12}$$

Whakaurua te \frac{89}{12} mō y ki x=-y+89. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{89}{12}+89$$

Tāpiri 89 ki te -\frac{89}{12}.

$$x=\frac{979}{12}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{979}{12},y=\frac{89}{12}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Tangohia te 23y mai i ngā taha e rua.

$$2x+y-23y=0$$

Pahekotia te y me -23y, ka -22y.

$$2x-22y=0$$

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

$$x+y=89,2x-22y=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\2&-22\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}89\\0\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&1\\2&-22\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&1\\2&-22\end{matrix}\right))\left(\begin{matrix}1&1\\2&-22\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\2&-22\end{matrix}\right))\left(\begin{matrix}89\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\2&-22\end{matrix}\right))\left(\begin{matrix}89\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\2&-22\end{matrix}\right))\left(\begin{matrix}89\\0\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{22}{-22-2}&-\frac{1}{-22-2}\\-\frac{2}{-22-2}&\frac{1}{-22-2}\end{matrix}\right)\left(\begin{matrix}89\\0\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{11}{12}&\frac{1}{24}\\\frac{1}{12}&-\frac{1}{24}\end{matrix}\right)\left(\begin{matrix}89\\0\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{11}{12}\times 89\\\frac{1}{12}\times 89\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{979}{12}\\\frac{89}{12}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{979}{12},y=\frac{89}{12}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Tangohia te 23y mai i ngā taha e rua.

$$2x+y-23y=0$$

Pahekotia te y me -23y, ka -22y.

$$2x-22y=0$$

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.

$$x+y=89,2x-22y=0$$

Kia ōrite ai a x me 2x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$2x+2y=2\times 89,2x-22y=0$$

Whakarūnātia.

$$2x+2y=178,2x-22y=0$$

Me tango 2x-22y=0 mai i 2x+2y=178 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2x-2x+2y+22y=178$$

Tāpiri 2x ki te -2x. Ka whakakore atu ngā kupu 2x me -2x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$2y+22y=178$$

Tāpiri 2y ki te 22y.

$$24y=178$$

Whakawehea ngā taha e rua ki te 24.

$$y=\frac{89}{12}$$

Whakaurua te \frac{89}{12} mō y ki 2x-22y=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x-22\times \frac{89}{12}=0$$

Whakareatia -22 ki te \frac{89}{12}.

$$2x-\frac{979}{6}=0$$

Me tāpiri \frac{979}{6} ki ngā taha e rua o te whārite.

$$2x=\frac{979}{6}$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{979}{12}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{979}{12},y=\frac{89}{12}$$