Whakaoti i te {l}{-9x-y=-14}{-x-5y=18}

$$\left. \begin{array} { l } { - 9 x - y = - 14 } \\ { - x - 5 y = 18 } \end{array} \right.$$

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.

$$-9x-y=-14,-x-5y=18$$

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

$$-9x-y=-14$$

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

$$-9x=y-14$$

Whakawehea ngā taha e rua ki te -9.

$$x=-\frac{1}{9}\left(y-14\right)$$

Whakareatia -\frac{1}{9} ki te y-14.

$$x=-\frac{1}{9}y+\frac{14}{9}$$

Whakakapia te \frac{-y+14}{9} mō te x ki tērā atu whārite, -x-5y=18.

$$-\left(-\frac{1}{9}y+\frac{14}{9}\right)-5y=18$$

Whakareatia -1 ki te \frac{-y+14}{9}.

$$\frac{1}{9}y-\frac{14}{9}-5y=18$$

Tāpiri \frac{y}{9} ki te -5y.

$$-\frac{44}{9}y-\frac{14}{9}=18$$

Me tāpiri \frac{14}{9} ki ngā taha e rua o te whārite.

$$-\frac{44}{9}y=\frac{176}{9}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{44}{9}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-4$$

Whakaurua te -4 mō y ki x=-\frac{1}{9}y+\frac{14}{9}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{9}\left(-4\right)+\frac{14}{9}$$

Whakareatia -\frac{1}{9} ki te -4.

$$x=\frac{4+14}{9}$$

Tāpiri \frac{14}{9} ki te \frac{4}{9} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=2$$

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.

$$-9x-y=-14,-x-5y=18$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-9&-1\\-1&-5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-14\\18\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-9&-1\\-1&-5\end{matrix}\right).

$$inverse(\left(\begin{matrix}-9&-1\\-1&-5\end{matrix}\right))\left(\begin{matrix}-9&-1\\-1&-5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-9&-1\\-1&-5\end{matrix}\right))\left(\begin{matrix}-14\\18\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-9&-1\\-1&-5\end{matrix}\right))\left(\begin{matrix}-14\\18\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-9&-1\\-1&-5\end{matrix}\right))\left(\begin{matrix}-14\\18\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{-9\left(-5\right)-\left(-\left(-1\right)\right)}&-\frac{-1}{-9\left(-5\right)-\left(-\left(-1\right)\right)}\\-\frac{-1}{-9\left(-5\right)-\left(-\left(-1\right)\right)}&-\frac{9}{-9\left(-5\right)-\left(-\left(-1\right)\right)}\end{matrix}\right)\left(\begin{matrix}-14\\18\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{44}&\frac{1}{44}\\\frac{1}{44}&-\frac{9}{44}\end{matrix}\right)\left(\begin{matrix}-14\\18\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{44}\left(-14\right)+\frac{1}{44}\times 18\\\frac{1}{44}\left(-14\right)-\frac{9}{44}\times 18\end{matrix}\right)$$

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.

$$-9x-y=-14,-x-5y=18$$

Kia ōrite ai a -9x 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 -9.

$$-\left(-9\right)x-\left(-y\right)=-\left(-14\right),-9\left(-1\right)x-9\left(-5\right)y=-9\times 18$$

Whakarūnātia.

$$9x+y=14,9x+45y=-162$$

Me tango 9x+45y=-162 mai i 9x+y=14 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$9x-9x+y-45y=14+162$$

Tāpiri 9x ki te -9x. Ka whakakore atu ngā kupu 9x me -9x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$y-45y=14+162$$

Tāpiri y ki te -45y.

$$-44y=14+162$$

Tāpiri 14 ki te 162.

$$-44y=176$$

Whakawehea ngā taha e rua ki te -44.

$$y=-4$$

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 tango 20 mai i ngā taha e rua o te whārite.

$$-x=-2$$

Whakawehea ngā taha e rua ki te -1.

$$x=2$$

Kua oti te pūnaha te whakatau.

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