Whakaoti i te {l}{20+x+y=115}{11x=8y}

$$\left\{ \begin{array} { l } { 20 + x + y = 115 } \\ { 11 x = 8 y } \end{array} \right.$$

Whakaoti mō x, y

$$x=40$$

$$y=55$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tangohia te 20 mai i ngā taha e rua.

$$x+y=115-20$$

Tangohia te 20 i te 115, ka 95.

$$x+y=95$$

Whakaarohia te whārite tuarua. Tangohia te 8y mai i ngā taha e rua.

$$11x-8y=0$$

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.

$$x+y=95,11x-8y=0$$

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

$$x+y=95$$

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

$$x=-y+95$$

Whakakapia te -y+95 mō te x ki tērā atu whārite, 11x-8y=0.

$$11\left(-y+95\right)-8y=0$$

Whakareatia 11 ki te -y+95.

$$-11y+1045-8y=0$$

Tāpiri -11y ki te -8y.

$$-19y+1045=0$$

Me tango 1045 mai i ngā taha e rua o te whārite.

$$-19y=-1045$$

Whakawehea ngā taha e rua ki te -19.

$$y=55$$

Whakaurua te 55 mō y ki x=-y+95. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-55+95$$

Tāpiri 95 ki te -55.

$$x=40$$

Kua oti te pūnaha te whakatau.

$$x=40,y=55$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tangohia te 20 mai i ngā taha e rua.

$$x+y=115-20$$

Tangohia te 20 i te 115, ka 95.

$$x+y=95$$

Whakaarohia te whārite tuarua. Tangohia te 8y mai i ngā taha e rua.

$$11x-8y=0$$

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

$$x+y=95,11x-8y=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\11&-8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}95\\0\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&1\\11&-8\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&1\\11&-8\end{matrix}\right))\left(\begin{matrix}1&1\\11&-8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\11&-8\end{matrix}\right))\left(\begin{matrix}95\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\11&-8\end{matrix}\right))\left(\begin{matrix}95\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\11&-8\end{matrix}\right))\left(\begin{matrix}95\\0\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{8}{-8-11}&-\frac{1}{-8-11}\\-\frac{11}{-8-11}&\frac{1}{-8-11}\end{matrix}\right)\left(\begin{matrix}95\\0\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{8}{19}&\frac{1}{19}\\\frac{11}{19}&-\frac{1}{19}\end{matrix}\right)\left(\begin{matrix}95\\0\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{8}{19}\times 95\\\frac{11}{19}\times 95\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}40\\55\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=40,y=55$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Tangohia te 20 mai i ngā taha e rua.

$$x+y=115-20$$

Tangohia te 20 i te 115, ka 95.

$$x+y=95$$

Whakaarohia te whārite tuarua. Tangohia te 8y mai i ngā taha e rua.

$$11x-8y=0$$

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.

$$x+y=95,11x-8y=0$$

Kia ōrite ai a x me 11x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 11 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$11x+11y=11\times 95,11x-8y=0$$

Whakarūnātia.

$$11x+11y=1045,11x-8y=0$$

Me tango 11x-8y=0 mai i 11x+11y=1045 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$11x-11x+11y+8y=1045$$

Tāpiri 11x ki te -11x. Ka whakakore atu ngā kupu 11x me -11x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$11y+8y=1045$$

Tāpiri 11y ki te 8y.

$$19y=1045$$

Whakawehea ngā taha e rua ki te 19.

$$y=55$$

Whakaurua te 55 mō y ki 11x-8y=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$11x-8\times 55=0$$

Whakareatia -8 ki te 55.

$$11x-440=0$$

Me tāpiri 440 ki ngā taha e rua o te whārite.

$$11x=440$$

Whakawehea ngā taha e rua ki te 11.

$$x=40$$

Kua oti te pūnaha te whakatau.

$$x=40,y=55$$