Whakaoti i te {l}{-7x-7y=14}{x+5y=-18}

$$\left\{ \begin{array}{l}{ - 7 x - 7 y = 14 }\\{ x + 5 y = - 18 }\end{array} \right.$$

Whakaoti mō x, y

$$x=2$$

$$y=-4$$

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.

$$-7x-7y=14,x+5y=-18$$

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

$$-7x-7y=14$$

Me tāpiri 7y ki ngā taha e rua o te whārite.

$$-7x=7y+14$$

Whakawehea ngā taha e rua ki te -7.

$$x=-\frac{1}{7}\left(7y+14\right)$$

Whakareatia -\frac{1}{7} ki te 14+7y.

$$x=-y-2$$

Whakakapia te -y-2 mō te x ki tērā atu whārite, x+5y=-18.

$$-y-2+5y=-18$$

Tāpiri -y ki te 5y.

$$4y-2=-18$$

Me tāpiri 2 ki ngā taha e rua o te whārite.

$$4y=-16$$

Whakawehea ngā taha e rua ki te 4.

$$y=-4$$

Whakaurua te -4 mō y ki x=-y-2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\left(-4\right)-2$$

Whakareatia -1 ki te -4.

$$x=4-2$$

Tāpiri -2 ki te 4.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=-4$$

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.

$$-7x-7y=14,x+5y=-18$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-7&-7\\1&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}14\\-18\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-7&-7\\1&5\end{matrix}\right).

$$inverse(\left(\begin{matrix}-7&-7\\1&5\end{matrix}\right))\left(\begin{matrix}-7&-7\\1&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-7&-7\\1&5\end{matrix}\right))\left(\begin{matrix}14\\-18\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-7&-7\\1&5\end{matrix}\right))\left(\begin{matrix}14\\-18\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-7&-7\\1&5\end{matrix}\right))\left(\begin{matrix}14\\-18\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{-7\times 5-\left(-7\right)}&-\frac{-7}{-7\times 5-\left(-7\right)}\\-\frac{1}{-7\times 5-\left(-7\right)}&-\frac{7}{-7\times 5-\left(-7\right)}\end{matrix}\right)\left(\begin{matrix}14\\-18\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{28}&-\frac{1}{4}\\\frac{1}{28}&\frac{1}{4}\end{matrix}\right)\left(\begin{matrix}14\\-18\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{28}\times 14-\frac{1}{4}\left(-18\right)\\\frac{1}{28}\times 14+\frac{1}{4}\left(-18\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2\\-4\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=2,y=-4$$

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.

$$-7x-7y=14,x+5y=-18$$

Kia ōrite ai a -7x me x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 1 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te -7.

$$-7x-7y=14,-7x-7\times 5y=-7\left(-18\right)$$

Whakarūnātia.

$$-7x-7y=14,-7x-35y=126$$

Me tango -7x-35y=126 mai i -7x-7y=14 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-7x+7x-7y+35y=14-126$$

Tāpiri -7x ki te 7x. Ka whakakore atu ngā kupu -7x me 7x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-7y+35y=14-126$$

Tāpiri -7y ki te 35y.

$$28y=14-126$$

Tāpiri 14 ki te -126.

$$28y=-112$$

Whakawehea ngā taha e rua ki te 28.

$$y=-4$$

Whakaurua te -4 mō y ki x+5y=-18. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x+5\left(-4\right)=-18$$

Whakareatia 5 ki te -4.

$$x-20=-18$$

Me tāpiri 20 ki ngā taha e rua o te whārite.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=-4$$