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{10}{3}+\frac{5y}{12} 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{10}{3}+\frac{5y}{12}.

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

Tāpiri 5y ki te -11y.

$$-6y+40=88$$

Me tango 40 mai i ngā taha e rua o te whārite.

$$-6y=48$$

Whakawehea ngā taha e rua ki te -6.

$$y=-8$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{11}{72}&-\frac{5}{72}\\\frac{1}{6}&-\frac{1}{6}\end{matrix}\right)\left(\begin{matrix}40\\88\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{11}{72}\times 40-\frac{5}{72}\times 88\\\frac{1}{6}\times 40-\frac{1}{6}\times 88\end{matrix}\right)$$

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.

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

Me tango 12x-11y=88 mai i 12x-5y=40 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

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

Tāpiri 12x ki te -12x. Ka whakakore atu ngā kupu 12x me -12x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

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

Tāpiri -5y ki te 11y.

$$6y=40-88$$

Tāpiri 40 ki te -88.

$$6y=-48$$

Whakawehea ngā taha e rua ki te 6.

$$y=-8$$

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.

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