Whakaoti i te {l}{3x-2y=0}{4x+y=5}

$$\left. \begin{array} { l } { 3 x - 2 y = 0 } \\ { 4 x + y = 5 } \end{array} \right.$$

Whakaoti mō x, y

$$x=\frac{10}{11}\approx 0.909090909$$

$$y = \frac{15}{11} = 1\frac{4}{11} \approx 1.363636364$$

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.

$$3x-2y=0,4x+y=5$$

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

$$3x-2y=0$$

Me tāpiri 2y ki ngā taha e rua o te whārite.

$$3x=2y$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\times 2y$$

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

$$x=\frac{2}{3}y$$

Whakakapia te \frac{2y}{3} mō te x ki tērā atu whārite, 4x+y=5.

$$4\times \frac{2}{3}y+y=5$$

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

$$\frac{8}{3}y+y=5$$

Tāpiri \frac{8y}{3} ki te y.

$$\frac{11}{3}y=5$$

Whakawehea ngā taha e rua o te whārite ki te \frac{11}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{15}{11}$$

Whakaurua te \frac{15}{11} mō y ki x=\frac{2}{3}y. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{2}{3}\times \frac{15}{11}$$

Whakareatia \frac{2}{3} ki te \frac{15}{11} 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.

$$x=\frac{10}{11}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{10}{11},y=\frac{15}{11}$$

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.

$$3x-2y=0,4x+y=5$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&-2\\4&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}0\\5\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&-2\\4&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&-2\\4&1\end{matrix}\right))\left(\begin{matrix}3&-2\\4&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-2\\4&1\end{matrix}\right))\left(\begin{matrix}0\\5\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-2\\4&1\end{matrix}\right))\left(\begin{matrix}0\\5\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-2\\4&1\end{matrix}\right))\left(\begin{matrix}0\\5\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3-\left(-2\times 4\right)}&-\frac{-2}{3-\left(-2\times 4\right)}\\-\frac{4}{3-\left(-2\times 4\right)}&\frac{3}{3-\left(-2\times 4\right)}\end{matrix}\right)\left(\begin{matrix}0\\5\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{11}&\frac{2}{11}\\-\frac{4}{11}&\frac{3}{11}\end{matrix}\right)\left(\begin{matrix}0\\5\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{11}\times 5\\\frac{3}{11}\times 5\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{10}{11}\\\frac{15}{11}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{10}{11},y=\frac{15}{11}$$

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.

$$3x-2y=0,4x+y=5$$

Kia ōrite ai a 3x me 4x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 4 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$4\times 3x+4\left(-2\right)y=0,3\times 4x+3y=3\times 5$$

Whakarūnātia.

$$12x-8y=0,12x+3y=15$$

Me tango 12x+3y=15 mai i 12x-8y=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$12x-12x-8y-3y=-15$$

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.

$$-8y-3y=-15$$

Tāpiri -8y ki te -3y.

$$-11y=-15$$

Whakawehea ngā taha e rua ki te -11.

$$y=\frac{15}{11}$$

Whakaurua te \frac{15}{11} mō y ki 4x+y=5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x+\frac{15}{11}=5$$

Me tango \frac{15}{11} mai i ngā taha e rua o te whārite.

$$4x=\frac{40}{11}$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{10}{11}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{10}{11},y=\frac{15}{11}$$