Whakaoti i te {l}{-12x-5y=40}{12x-11y=88}

$$\left. \begin{array} { l } { - 12 x - 5 y = 40 } \\ { 12 x - 11 y = 88 } \end{array} \right.$$

Whakaoti mō x, y

$$x=0$$

$$y=-8$$

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.

$$-12x-5y=40,12x-11y=88$$

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

$$-12x-5y=40$$

Me tāpiri 5y ki ngā taha e rua o te whārite.

$$-12x=5y+40$$

Whakawehea ngā taha e rua ki te -12.

$$x=-\frac{1}{12}\left(5y+40\right)$$

Whakareatia -\frac{1}{12} ki te 40+5y.

$$x=-\frac{5}{12}y-\frac{10}{3}$$

Whakakapia te -\frac{5y}{12}-\frac{10}{3} mō te x ki tērā atu whārite, 12x-11y=88.

$$12\left(-\frac{5}{12}y-\frac{10}{3}\right)-11y=88$$

Whakareatia 12 ki te -\frac{5y}{12}-\frac{10}{3}.

$$-5y-40-11y=88$$

Tāpiri -5y ki te -11y.

$$-16y-40=88$$

Me tāpiri 40 ki ngā taha e rua o te whārite.

$$-16y=128$$

Whakawehea ngā taha e rua ki te -16.

$$y=-8$$

Whakaurua te -8 mō y ki x=-\frac{5}{12}y-\frac{10}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{5}{12}\left(-8\right)-\frac{10}{3}$$

Whakareatia -\frac{5}{12} ki te -8.

$$x=\frac{10-10}{3}$$

Tāpiri -\frac{10}{3} ki te \frac{10}{3} 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.

$$x=0$$

Kua oti te pūnaha te whakatau.

$$x=0,y=-8$$

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.

$$-12x-5y=40,12x-11y=88$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-12&-5\\12&-11\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}40\\88\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-12&-5\\12&-11\end{matrix}\right).

$$inverse(\left(\begin{matrix}-12&-5\\12&-11\end{matrix}\right))\left(\begin{matrix}-12&-5\\12&-11\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-12&-5\\12&-11\end{matrix}\right))\left(\begin{matrix}40\\88\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-12&-5\\12&-11\end{matrix}\right))\left(\begin{matrix}40\\88\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-12&-5\\12&-11\end{matrix}\right))\left(\begin{matrix}40\\88\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{11}{-12\left(-11\right)-\left(-5\times 12\right)}&-\frac{-5}{-12\left(-11\right)-\left(-5\times 12\right)}\\-\frac{12}{-12\left(-11\right)-\left(-5\times 12\right)}&-\frac{12}{-12\left(-11\right)-\left(-5\times 12\right)}\end{matrix}\right)\left(\begin{matrix}40\\88\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{11}{192}&\frac{5}{192}\\-\frac{1}{16}&-\frac{1}{16}\end{matrix}\right)\left(\begin{matrix}40\\88\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{11}{192}\times 40+\frac{5}{192}\times 88\\-\frac{1}{16}\times 40-\frac{1}{16}\times 88\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=0,y=-8$$

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.

$$-12x-5y=40,12x-11y=88$$

Kia ōrite ai a -12x me 12x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 12 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te -12.

$$12\left(-12\right)x+12\left(-5\right)y=12\times 40,-12\times 12x-12\left(-11\right)y=-12\times 88$$

Whakarūnātia.

$$-144x-60y=480,-144x+132y=-1056$$

Me tango -144x+132y=-1056 mai i -144x-60y=480 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-144x+144x-60y-132y=480+1056$$

Tāpiri -144x ki te 144x. Ka whakakore atu ngā kupu -144x me 144x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-60y-132y=480+1056$$

Tāpiri -60y ki te -132y.

$$-192y=480+1056$$

Tāpiri 480 ki te 1056.

$$-192y=1536$$

Whakawehea ngā taha e rua ki te -192.

$$y=-8$$

Whakaurua te -8 mō y ki 12x-11y=88. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$12x-11\left(-8\right)=88$$

Whakareatia -11 ki te -8.

$$12x+88=88$$

Me tango 88 mai i ngā taha e rua o te whārite.

$$12x=0$$

Whakawehea ngā taha e rua ki te 12.

$$x=0$$

Kua oti te pūnaha te whakatau.

$$x=0,y=-8$$