Whakaoti i te 3x+2y=12,12=9x-8y

$$3 x + 2 y = 12,12 = 9 x - 8 y$$

Whakaoti mō x, y

$$x = \frac{20}{7} = 2\frac{6}{7} \approx 2.857142857$$

$$y = \frac{12}{7} = 1\frac{5}{7} \approx 1.714285714$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$9x-8y=12$$

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.

$$3x+2y=12,9x-8y=12$$

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

$$3x+2y=12$$

Me tango 2y mai i ngā taha e rua o te whārite.

$$3x=-2y+12$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(-2y+12\right)$$

Whakareatia \frac{1}{3} ki te -2y+12.

$$x=-\frac{2}{3}y+4$$

Whakakapia te -\frac{2y}{3}+4 mō te x ki tērā atu whārite, 9x-8y=12.

$$9\left(-\frac{2}{3}y+4\right)-8y=12$$

Whakareatia 9 ki te -\frac{2y}{3}+4.

$$-6y+36-8y=12$$

Tāpiri -6y ki te -8y.

$$-14y+36=12$$

Me tango 36 mai i ngā taha e rua o te whārite.

$$-14y=-24$$

Whakawehea ngā taha e rua ki te -14.

$$y=\frac{12}{7}$$

Whakaurua te \frac{12}{7} mō y ki x=-\frac{2}{3}y+4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{2}{3}\times \frac{12}{7}+4$$

Whakareatia -\frac{2}{3} ki te \frac{12}{7} 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.

$$x=-\frac{8}{7}+4$$

Tāpiri 4 ki te -\frac{8}{7}.

$$x=\frac{20}{7}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{20}{7},y=\frac{12}{7}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$9x-8y=12$$

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$3x+2y=12,9x-8y=12$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&2\\9&-8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}12\\12\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&2\\9&-8\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&2\\9&-8\end{matrix}\right))\left(\begin{matrix}3&2\\9&-8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&2\\9&-8\end{matrix}\right))\left(\begin{matrix}12\\12\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&2\\9&-8\end{matrix}\right))\left(\begin{matrix}12\\12\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&2\\9&-8\end{matrix}\right))\left(\begin{matrix}12\\12\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{8}{3\left(-8\right)-2\times 9}&-\frac{2}{3\left(-8\right)-2\times 9}\\-\frac{9}{3\left(-8\right)-2\times 9}&\frac{3}{3\left(-8\right)-2\times 9}\end{matrix}\right)\left(\begin{matrix}12\\12\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{21}&\frac{1}{21}\\\frac{3}{14}&-\frac{1}{14}\end{matrix}\right)\left(\begin{matrix}12\\12\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{21}\times 12+\frac{1}{21}\times 12\\\frac{3}{14}\times 12-\frac{1}{14}\times 12\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{20}{7}\\\frac{12}{7}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{20}{7},y=\frac{12}{7}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$9x-8y=12$$

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.

$$3x+2y=12,9x-8y=12$$

Kia ōrite ai a 3x me 9x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 9 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$9\times 3x+9\times 2y=9\times 12,3\times 9x+3\left(-8\right)y=3\times 12$$

Whakarūnātia.

$$27x+18y=108,27x-24y=36$$

Me tango 27x-24y=36 mai i 27x+18y=108 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$27x-27x+18y+24y=108-36$$

Tāpiri 27x ki te -27x. Ka whakakore atu ngā kupu 27x me -27x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$18y+24y=108-36$$

Tāpiri 18y ki te 24y.

$$42y=108-36$$

Tāpiri 108 ki te -36.

$$42y=72$$

Whakawehea ngā taha e rua ki te 42.

$$y=\frac{12}{7}$$

Whakaurua te \frac{12}{7} mō y ki 9x-8y=12. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$9x-8\times \frac{12}{7}=12$$

Whakareatia -8 ki te \frac{12}{7}.

$$9x-\frac{96}{7}=12$$

Me tāpiri \frac{96}{7} ki ngā taha e rua o te whārite.

$$9x=\frac{180}{7}$$

Whakawehea ngā taha e rua ki te 9.

$$x=\frac{20}{7}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{20}{7},y=\frac{12}{7}$$