Whakaoti i te {l}{x+y=72}{x=63y}

$$\left. \begin{array} { l } { x + y = 72 } \\ { x = 63 y } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{567}{8} = 70\frac{7}{8} = 70.875$$

$$y = \frac{9}{8} = 1\frac{1}{8} = 1.125$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Tangohia te 63y mai i ngā taha e rua.

$$x-63y=0$$

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.

$$x+y=72,x-63y=0$$

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

$$x+y=72$$

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

$$x=-y+72$$

Whakakapia te -y+72 mō te x ki tērā atu whārite, x-63y=0.

$$-y+72-63y=0$$

Tāpiri -y ki te -63y.

$$-64y+72=0$$

Me tango 72 mai i ngā taha e rua o te whārite.

$$-64y=-72$$

Whakawehea ngā taha e rua ki te -64.

$$y=\frac{9}{8}$$

Whakaurua te \frac{9}{8} mō y ki x=-y+72. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{9}{8}+72$$

Tāpiri 72 ki te -\frac{9}{8}.

$$x=\frac{567}{8}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{567}{8},y=\frac{9}{8}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Tangohia te 63y mai i ngā taha e rua.

$$x-63y=0$$

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

$$x+y=72,x-63y=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\1&-63\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}72\\0\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&1\\1&-63\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&1\\1&-63\end{matrix}\right))\left(\begin{matrix}1&1\\1&-63\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\1&-63\end{matrix}\right))\left(\begin{matrix}72\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\1&-63\end{matrix}\right))\left(\begin{matrix}72\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\1&-63\end{matrix}\right))\left(\begin{matrix}72\\0\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{63}{-63-1}&-\frac{1}{-63-1}\\-\frac{1}{-63-1}&\frac{1}{-63-1}\end{matrix}\right)\left(\begin{matrix}72\\0\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{63}{64}&\frac{1}{64}\\\frac{1}{64}&-\frac{1}{64}\end{matrix}\right)\left(\begin{matrix}72\\0\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{63}{64}\times 72\\\frac{1}{64}\times 72\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{567}{8}\\\frac{9}{8}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{567}{8},y=\frac{9}{8}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Tangohia te 63y mai i ngā taha e rua.

$$x-63y=0$$

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.

$$x+y=72,x-63y=0$$

Me tango x-63y=0 mai i x+y=72 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$x-x+y+63y=72$$

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

$$y+63y=72$$

Tāpiri y ki te 63y.

$$64y=72$$

Whakawehea ngā taha e rua ki te 64.

$$y=\frac{9}{8}$$

Whakaurua te \frac{9}{8} mō y ki x-63y=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x-63\times \frac{9}{8}=0$$

Whakareatia -63 ki te \frac{9}{8}.

$$x-\frac{567}{8}=0$$

Me tāpiri \frac{567}{8} ki ngā taha e rua o te whārite.

$$x=\frac{567}{8}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{567}{8},y=\frac{9}{8}$$