Whakaoti i te {l}{22x+y=50}{27x-y=96}

$$\left. \begin{array} { l } { 22 x + y = 50 } \\ { 27 x - y = 96 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{146}{49} = 2\frac{48}{49} \approx 2.979591837$$

$$y = -\frac{762}{49} = -15\frac{27}{49} \approx -15.551020408$$

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.

$$22x+y=50,27x-y=96$$

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

$$22x+y=50$$

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

$$22x=-y+50$$

Whakawehea ngā taha e rua ki te 22.

$$x=\frac{1}{22}\left(-y+50\right)$$

Whakareatia \frac{1}{22} ki te -y+50.

$$x=-\frac{1}{22}y+\frac{25}{11}$$

Whakakapia te -\frac{y}{22}+\frac{25}{11} mō te x ki tērā atu whārite, 27x-y=96.

$$27\left(-\frac{1}{22}y+\frac{25}{11}\right)-y=96$$

Whakareatia 27 ki te -\frac{y}{22}+\frac{25}{11}.

$$-\frac{27}{22}y+\frac{675}{11}-y=96$$

Tāpiri -\frac{27y}{22} ki te -y.

$$-\frac{49}{22}y+\frac{675}{11}=96$$

Me tango \frac{675}{11} mai i ngā taha e rua o te whārite.

$$-\frac{49}{22}y=\frac{381}{11}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{49}{22}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-\frac{762}{49}$$

Whakaurua te -\frac{762}{49} mō y ki x=-\frac{1}{22}y+\frac{25}{11}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{22}\left(-\frac{762}{49}\right)+\frac{25}{11}$$

Whakareatia -\frac{1}{22} ki te -\frac{762}{49} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=\frac{381}{539}+\frac{25}{11}$$

Tāpiri \frac{25}{11} ki te \frac{381}{539} 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.

$$x=\frac{146}{49}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{146}{49},y=-\frac{762}{49}$$

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.

$$22x+y=50,27x-y=96$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}22&1\\27&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}50\\96\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}22&1\\27&-1\end{matrix}\right).

$$inverse(\left(\begin{matrix}22&1\\27&-1\end{matrix}\right))\left(\begin{matrix}22&1\\27&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}22&1\\27&-1\end{matrix}\right))\left(\begin{matrix}50\\96\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}22&1\\27&-1\end{matrix}\right))\left(\begin{matrix}50\\96\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}22&1\\27&-1\end{matrix}\right))\left(\begin{matrix}50\\96\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{22\left(-1\right)-27}&-\frac{1}{22\left(-1\right)-27}\\-\frac{27}{22\left(-1\right)-27}&\frac{22}{22\left(-1\right)-27}\end{matrix}\right)\left(\begin{matrix}50\\96\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{49}&\frac{1}{49}\\\frac{27}{49}&-\frac{22}{49}\end{matrix}\right)\left(\begin{matrix}50\\96\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{49}\times 50+\frac{1}{49}\times 96\\\frac{27}{49}\times 50-\frac{22}{49}\times 96\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{146}{49}\\-\frac{762}{49}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{146}{49},y=-\frac{762}{49}$$

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.

$$22x+y=50,27x-y=96$$

Kia ōrite ai a 22x me 27x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 27 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 22.

$$27\times 22x+27y=27\times 50,22\times 27x+22\left(-1\right)y=22\times 96$$

Whakarūnātia.

$$594x+27y=1350,594x-22y=2112$$

Me tango 594x-22y=2112 mai i 594x+27y=1350 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$594x-594x+27y+22y=1350-2112$$

Tāpiri 594x ki te -594x. Ka whakakore atu ngā kupu 594x me -594x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$27y+22y=1350-2112$$

Tāpiri 27y ki te 22y.

$$49y=1350-2112$$

Tāpiri 1350 ki te -2112.

$$49y=-762$$

Whakawehea ngā taha e rua ki te 49.

$$y=-\frac{762}{49}$$

Whakaurua te -\frac{762}{49} mō y ki 27x-y=96. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$27x-\left(-\frac{762}{49}\right)=96$$

Me tango \frac{762}{49} mai i ngā taha e rua o te whārite.

$$27x=\frac{3942}{49}$$

Whakawehea ngā taha e rua ki te 27.

$$x=\frac{146}{49}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{146}{49},y=-\frac{762}{49}$$