Whakaoti i te {l}{x+y=140}{x/y=7}

$$\left\{ \begin{array} { l } { x + y = 140 } \\ { \frac { x } { y } = 7 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{245}{2} = 122\frac{1}{2} = 122.5$$

$$y = \frac{35}{2} = 17\frac{1}{2} = 17.5$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Tē taea kia ōrite te tāupe y ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te y.

$$x=7y$$

Tangohia te 7y mai i ngā taha e rua.

$$x-7y=0$$

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.

$$x+y=140,x-7y=0$$

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

$$x+y=140$$

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

$$x=-y+140$$

Whakakapia te -y+140 mō te x ki tērā atu whārite, x-7y=0.

$$-y+140-7y=0$$

Tāpiri -y ki te -7y.

$$-8y+140=0$$

Me tango 140 mai i ngā taha e rua o te whārite.

$$-8y=-140$$

Whakawehea ngā taha e rua ki te -8.

$$y=\frac{35}{2}$$

Whakaurua te \frac{35}{2} mō y ki x=-y+140. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{35}{2}+140$$

Tāpiri 140 ki te -\frac{35}{2}.

$$x=\frac{245}{2}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{245}{2},y=\frac{35}{2}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Tē taea kia ōrite te tāupe y ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te y.

$$x=7y$$

Tangohia te 7y mai i ngā taha e rua.

$$x-7y=0$$

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

$$x+y=140,x-7y=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\1&-7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}140\\0\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&1\\1&-7\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&1\\1&-7\end{matrix}\right))\left(\begin{matrix}1&1\\1&-7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\1&-7\end{matrix}\right))\left(\begin{matrix}140\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\1&-7\end{matrix}\right))\left(\begin{matrix}140\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\1&-7\end{matrix}\right))\left(\begin{matrix}140\\0\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{7}{-7-1}&-\frac{1}{-7-1}\\-\frac{1}{-7-1}&\frac{1}{-7-1}\end{matrix}\right)\left(\begin{matrix}140\\0\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{8}&\frac{1}{8}\\\frac{1}{8}&-\frac{1}{8}\end{matrix}\right)\left(\begin{matrix}140\\0\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{8}\times 140\\\frac{1}{8}\times 140\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{245}{2}\\\frac{35}{2}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{245}{2},y=\frac{35}{2}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Tē taea kia ōrite te tāupe y ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te y.

$$x=7y$$

Tangohia te 7y mai i ngā taha e rua.

$$x-7y=0$$

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.

$$x+y=140,x-7y=0$$

Me tango x-7y=0 mai i x+y=140 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$x-x+y+7y=140$$

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.

$$y+7y=140$$

Tāpiri y ki te 7y.

$$8y=140$$

Whakawehea ngā taha e rua ki te 8.

$$y=\frac{35}{2}$$

Whakaurua te \frac{35}{2} mō y ki x-7y=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x-7\times \frac{35}{2}=0$$

Whakareatia -7 ki te \frac{35}{2}.

$$x-\frac{245}{2}=0$$

Me tāpiri \frac{245}{2} ki ngā taha e rua o te whārite.

$$x=\frac{245}{2}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{245}{2},y=\frac{35}{2}$$