Whakaoti i te {l}{x+y=69}{2x+y=87}

$$\left. \begin{array} { l } { x + y = 69 } \\ { 2 x + y = 87 } \end{array} \right.$$

Whakaoti mō x, y

$$x=18$$

$$y=51$$

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.

$$x+y=69,2x+y=87$$

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

$$x+y=69$$

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

$$x=-y+69$$

Whakakapia te -y+69 mō te x ki tērā atu whārite, 2x+y=87.

$$2\left(-y+69\right)+y=87$$

Whakareatia 2 ki te -y+69.

$$-2y+138+y=87$$

Tāpiri -2y ki te y.

$$-y+138=87$$

Me tango 138 mai i ngā taha e rua o te whārite.

$$-y=-51$$

Whakawehea ngā taha e rua ki te -1.

$$y=51$$

Whakaurua te 51 mō y ki x=-y+69. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-51+69$$

Tāpiri 69 ki te -51.

$$x=18$$

Kua oti te pūnaha te whakatau.

$$x=18,y=51$$

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.

$$x+y=69,2x+y=87$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\2&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}69\\87\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&1\\2&1\end{matrix}\right))\left(\begin{matrix}1&1\\2&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\2&1\end{matrix}\right))\left(\begin{matrix}69\\87\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\2&1\end{matrix}\right))\left(\begin{matrix}69\\87\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\2&1\end{matrix}\right))\left(\begin{matrix}69\\87\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{1-2}&-\frac{1}{1-2}\\-\frac{2}{1-2}&\frac{1}{1-2}\end{matrix}\right)\left(\begin{matrix}69\\87\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-1&1\\2&-1\end{matrix}\right)\left(\begin{matrix}69\\87\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-69+87\\2\times 69-87\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}18\\51\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=18,y=51$$

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.

$$x+y=69,2x+y=87$$

Me tango 2x+y=87 mai i x+y=69 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$x-2x+y-y=69-87$$

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

$$x-2x=69-87$$

Tāpiri x ki te -2x.

$$-x=69-87$$

Tāpiri 69 ki te -87.

$$-x=-18$$

Whakawehea ngā taha e rua ki te -1.

$$x=18$$

Whakaurua te 18 mō x ki 2x+y=87. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$2\times 18+y=87$$

Whakareatia 2 ki te 18.

$$36+y=87$$

Me tango 36 mai i ngā taha e rua o te whārite.

$$y=51$$

Kua oti te pūnaha te whakatau.

$$x=18,y=51$$