Whakaoti i te {l}{4x+2y=12}{7x+18y=19}

$$\left. \begin{array} { l } { 4 x + 2 y = 12 } \\ { 7 x + 18 y = 19 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{89}{29} = 3\frac{2}{29} \approx 3.068965517$$

$$y=-\frac{4}{29}\approx -0.137931034$$

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.

$$4x+2y=12,7x+18y=19$$

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

$$4x+2y=12$$

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

$$4x=-2y+12$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{1}{4}\left(-2y+12\right)$$

Whakareatia \frac{1}{4} ki te -2y+12.

$$x=-\frac{1}{2}y+3$$

Whakakapia te -\frac{y}{2}+3 mō te x ki tērā atu whārite, 7x+18y=19.

$$7\left(-\frac{1}{2}y+3\right)+18y=19$$

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

$$-\frac{7}{2}y+21+18y=19$$

Tāpiri -\frac{7y}{2} ki te 18y.

$$\frac{29}{2}y+21=19$$

Me tango 21 mai i ngā taha e rua o te whārite.

$$\frac{29}{2}y=-2$$

Whakawehea ngā taha e rua o te whārite ki te \frac{29}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-\frac{4}{29}$$

Whakaurua te -\frac{4}{29} mō y ki x=-\frac{1}{2}y+3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{2}\left(-\frac{4}{29}\right)+3$$

Whakareatia -\frac{1}{2} ki te -\frac{4}{29} 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.

$$x=\frac{2}{29}+3$$

Tāpiri 3 ki te \frac{2}{29}.

$$x=\frac{89}{29}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{89}{29},y=-\frac{4}{29}$$

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.

$$4x+2y=12,7x+18y=19$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&2\\7&18\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}12\\19\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}4&2\\7&18\end{matrix}\right).

$$inverse(\left(\begin{matrix}4&2\\7&18\end{matrix}\right))\left(\begin{matrix}4&2\\7&18\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&2\\7&18\end{matrix}\right))\left(\begin{matrix}12\\19\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&2\\7&18\end{matrix}\right))\left(\begin{matrix}12\\19\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&2\\7&18\end{matrix}\right))\left(\begin{matrix}12\\19\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te poukapa kōaro ko \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), kia tuhia anō ai te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{18}{4\times 18-2\times 7}&-\frac{2}{4\times 18-2\times 7}\\-\frac{7}{4\times 18-2\times 7}&\frac{4}{4\times 18-2\times 7}\end{matrix}\right)\left(\begin{matrix}12\\19\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{9}{29}&-\frac{1}{29}\\-\frac{7}{58}&\frac{2}{29}\end{matrix}\right)\left(\begin{matrix}12\\19\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{9}{29}\times 12-\frac{1}{29}\times 19\\-\frac{7}{58}\times 12+\frac{2}{29}\times 19\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{89}{29}\\-\frac{4}{29}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{89}{29},y=-\frac{4}{29}$$

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.

$$4x+2y=12,7x+18y=19$$

Kia ōrite ai a 4x me 7x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 7 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 4.

$$7\times 4x+7\times 2y=7\times 12,4\times 7x+4\times 18y=4\times 19$$

Whakarūnātia.

$$28x+14y=84,28x+72y=76$$

Me tango 28x+72y=76 mai i 28x+14y=84 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$28x-28x+14y-72y=84-76$$

Tāpiri 28x ki te -28x. Ka whakakore atu ngā kupu 28x me -28x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$14y-72y=84-76$$

Tāpiri 14y ki te -72y.

$$-58y=84-76$$

Tāpiri 84 ki te -76.

$$-58y=8$$

Whakawehea ngā taha e rua ki te -58.

$$y=-\frac{4}{29}$$

Whakaurua te -\frac{4}{29} mō y ki 7x+18y=19. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$7x+18\left(-\frac{4}{29}\right)=19$$

Whakareatia 18 ki te -\frac{4}{29}.

$$7x-\frac{72}{29}=19$$

Me tāpiri \frac{72}{29} ki ngā taha e rua o te whārite.

$$7x=\frac{623}{29}$$

Whakawehea ngā taha e rua ki te 7.

$$x=\frac{89}{29}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{89}{29},y=-\frac{4}{29}$$